Post-Wick theorems for symbolic manipulation of second-quantized expressions

Manipulating expressions in many-body perturbation theory becomes unwieldy with increasing order of the perturbation theory. Here I derive a set of theorems for efficient simplification of such expressions. The derived rules are specifically designed for implementing with symbolic algebra tools. As an illustration, we count the numbers of Brueckner–Goldstone diagrams in the first several orders of many-body perturbation theory for matrix elements between two states of a mono-valent system. Many-body perturbation theory (MBPT) has proven to be a powerful tool in physics [1] and quantum chemistry [2]. Although MBPT provides a systematic approach to solving the many-body problem, the number and complexity of analytical expressions becomes rapidly unwieldy with increasing order of perturbation theory. At the same time, exploring higher orders is desirable for improving accuracy of ab initio atomicstructure methods. Here, a number of applications may benefit, ranging from atomic parity violation [3] and atomic clocks [4, 5] to a precision characterization of long-range inter-atomic potentials for ultra-cold collision studies [6]. To overcome an overwhelming complexity of the MBPT in high orders, one has to develop symbolic tools that automate highly repetitive but error-prone derivation of manybody diagrams. The advantage of using symbolic algebra computing for these goals has been realized for a number of decades. For example, the pioneering ‘Schoonschip’ program [7] and other symbolic packages are employed for evaluating Feynman diagrams in quantum electrodynamics and highenergy physics. We also note similar efforts in quantum chemistry [8] (see also [9] and references therein). In atomic MBPT, developing symbolic tools was reported by the Notre Dame [10], Michigan [11], and very recently by the Sydney [12] and Kassel groups [13]. In practical applications of MBPT, one deals with products of strings of creation and annihilation operators. Typically such products are evaluated with Wick’s theorem (see, e.g. discussion in [14]). This is the point of departure of the symbolic calculations described in [10] and [11]. The application of Wick’s theorem results in a series of Kronecker delta symbols. The next step in the derivation requires carrying out summation over the delta symbols. In a typical application, the resulting terms are redundant and require additional efforts to further simplify and combine the expressions. The complexity of both the application of Wick’s theorem and the further simplification grows rapidly as the order of perturbation theory increases. Over the past decade, our group in Reno has developed an alternative set of symbolic tools for MBPT. The goal of our work was to study high orders of MBPT, e.g. fourthorder contributions to matrix elements for mono-valent atoms [15, 16]. In our practical work, we found that the conventional approaches based on the straightforward applications of Wick’s theorem require prohibitively long computational times. To overcome this difficulty, I have derived a set of rules enabling efficient derivation of MBPT expressions for fermionic systems in high orders of MBPT. These theorems are reported here. Wick’s theorem works at the level of elemental pairwise contractions of creation and annihilation operators. The basic idea of the present approach is to shortcut directly to the resulting expressions for typical operations in MBPT, without the need to apply expensive pairwise operations. The theorems are formulated as a set of symbolic replacement 0953-4075/10/074001+09$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK & the USA J. Phys. B: At. Mol. Opt. Phys. 43 (2010) 074001 A Derevianko rules, ideally suited for implementing with symbolic algebra systems. We provide an accompanying Mathematica package downloadable from the author’s website [17]. In this work, we focus on mono-valent systems. The paper is organized as follows. In section 1, we review main results from the many-body perturbation theory and introduce notation. In sections 2.1 and 2.3, we derive rules for multiplying second-quantized operators with atomic wavefunctions. Similar theorems are derived for determining MBPT corrections to energies and matrix elements in section 3. Finally, as an illustration, in section 4 we derive explicit expressions for matrix elements in several first orders of MBPT and count the number of resulting diagrams. 1. Background and notation 1.1. Second quantization, normal forms and Wick’s theorems We start by recapitulating relevant notation and results from the second-quantization method as applied to fermionic systems. At the heart of the second quantization technique lies an expansion of the true many-body wave function over properly anti-symmetrized products of single-particle orbitals (the Slater determinants). The machinery is simplified by introducing the creation ( a † k ) and the annihilation (ak) operators satisfying the anti-commutation relations a † j a † k = −a ka j , ajak = −akaj , aja † k = δjk − a kaj , ajaj ≡ 0, a j a j ≡ 0. Applying strings of creation and annihilation operators to the vacuum state |0〉 builds the Slater determinants. A one-particle operator in the second quantization (such as an interaction with an external field) reads Z = ∑