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Wick : A Tool for Symbolic Manipulation of Creation and Annihilation Operators in Fermi Statistics

A. Derevianko Department of Physics, Notre Dame University, Notre Dame, IN 46556 (May 22, 1997)

Abstract The Mathematica package for symbolic manipulations of expressions involving creation and annihilation operators in Fermi statistics is presented. The description of the package with an example of derivation of all-order equations for an atomic system with closed core from Ref. [3] are given. Keywords: Fermi statistics; Second Quantization; Many-Body Perturbation Theory; Symbolic Computation PACS Codes: 31.15.-p, 31.15.Md

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Author Contact Info:

Andrei Derevianko Department of Physics, Notre Dame University, Notre Dame, IN 46556 USA email : [email protected] phone : (219) 631-6590 FAX : (219) 631-5952

Section of the Program Library Index : Atomic Physics.Theoretical Methods

PROGRAM SUMMARY

Title of program: Wick Catalogue identi er: Program obtainable from: CPC Program Library, Queen's University of Belfast, N. lreland Computer for which the program is designed and others on which it has been tested: The package is written in the Mathematica programming language and can be used on any computer supporting the Mathematica system. Operating systems or monitors under which the program has been tested: IBM RS/6000 running Mathematica 2.2 under AIX v4.1 Programming language used: Mathematica No. of bytes in distributed program, including test data, etc.: 23842 Distribution format: ASCII

2

Keywords: Fermi statistics; Second Quantization; Many-Body Perturbation Theory; Symbolic Computation Nature of physical problem The package provides a set of Mathematica functions to symbolically simplify the expressions involving creation and annihilation operators in Fermi statistics de ned with respect to a core vacuum state Method of solution The solution is based on anticommutation relations between operators, de nition of normal form of operator products, and Wick theorem Typical running time Depends greatly on the number of creation and annihilation operators involved

LONG WRITE-UP

I. INTRODUCTION

The conventional approach in atomic structure theory calculations is to use creation and annihilation operators in Fermi statistics. Such a formalism allows one to handle Slater determinants transparently. However, when the number of operators grows, the derivation of expressions becomes rather tedious and error-prone. On the other hand the underlying manipulation rules are quite straightforward and could be implemented with symbolic systems. For instance, the third order perturbative expressions in [1] were partially obtained using a package developed with Reduce. Unfortunately, this package has not been documented and made available to the atomic structure community. 3

This paper describes a Mathematica [2] package called Wick for manipulating expressions containing creation and annihilation operators in Fermi statistics. As an example, we reproduce the all-order equations of Many Body Perturbation Theory (MBPT) for a closed core atomic system from Ref. [3]. Also, we present an alternative way of formulation of all-order equations in antisymmetrized form. II.

PROBLEM SETUP

The creation ayi and annihilation aj operators in Fermi statistics obey the following anticommutation relations: = ayk ayj ; aj ak = ak aj ; y y aj ak = Æjk ak aj y y

aj ak

(2.1)

The action on vacuum state joi is de ned as aijoi = 0. This rule together with anticommutators to a set of simpli cation rules in the case of pure vacuum. The goal of simpli cation is to remove annihilation operators from the expressions with joi, and creation operators from expressions with hoj. When dealing with complex atoms one introduces core vacuum state joci and explicitly separates one-particle states into classes of excited (virtual) and core orbitals. The anticommutation relations Eqns. 2.1 are still valid, with an obvious condition that the Kronecker symbol of core and virtual orbitals vanishes. Additionally we have :

j i = 0; avirt joci = 0 ; hocjacore = 0; hocjayvirt = 0:

aycore oc

(2.2)

Evidently, the vacuum joi problem is a subclass of more general core joci case. In order to obtain the vacuum joi result one has to work in terms of virtual orbitals. The package employs the scheme of Ref. [4] to label classes of orbitals. In this convention the letters in the beginning of alphabet a; b; c; d span the indexes of core orbitals, letters in 4

the middle m; n; r; s; t represent excited orbitals, and the valence orbitals are labeled with v and w . Indexes i; j; k and l are reserved for labeling any of the above orbitals. For example, the wave-function of a system with one valence electron outside a closed core in the independent particle approximation is represented as ayv joci. The physical operators are expressed in terms of operators in normal form : A :. The normal form of operator is obtained by rearranging the operators contained in A so that y acore and avirt appear to the left of aycore and avirt . The sign of such the normal product is given by by the signature of permutation sequence. The zeroth-order independent particle Hamiltonian H0, one-particle (U ) and two-particle (G) operators are de ned as X

H0

=

U

=

G

= 21

k

X kl

:

:

y

k ak ak ;

:

:

y

(2.3)

ukl ak al ;

X ijkl

:

y y

:

gijkl ai aj al ak :

The total Hamiltonian also contains core energy Ec. The package employs an explicit expansion of normal forms into sums over core and excited orbitals, as discussed in Ref. [1]. For example, : aiayj : =

X ab

y

Æai Æbj aa ab

+

X am

X

Æai Æjm aa aym

am

Æaj Æim aya am

X mn

Æim Æjn ayn am :

(2.4)

After that, the anticommutation rules together with the de ned actions on the core are employed to simplify the expressions. The simpli ed output is represented in normal form. The following selection rules, based on Wick's theorem [4] are used to speed up the evaluation of matrix elements h0cjAj0ci. Such matrix elements vanish if 1. The number of ayvirt 6= the number of avirt , 2. The number of aycore 6= the number of acore. In general, the operator A could be represented as a product of three operators in normal form: A = : A1 :: A2 :: A3 :. The matrix elements give non-zero value only when the numbers 5

of operators in the corresponding forms : Ai : satisfy a triangular condition jl1 l2j  l3  y y l1 + l2 . For example, the matrix element h0c jav : ai aj al ak :ayv joc i vanishes since it does not satisfy the above rule. In addition, only one term of the Wick expansion of A into normal products contributes to the value of the matrix element | the term with the maximum possible number of contractions. li

III.

PACKAGE FUNCTIONS

The package is loaded with a command