Zombie states for description of structure and dynamics of multi-electron systems.

Dmitrii V Shalashilin

doi:10.1063/1.5023209

Abstract

Canonical Coherent States (CSs) of Harmonic Oscillator have been extensively used as a basis in a number of computational methods of quantum dynamics. However, generalising such techniques for fermionic systems is difficult because Fermionic Coherent States (FCSs) require complicated algebra of Grassmann numbers not well suited for numerical calculations. This paper introduces a coherent antisymmetrised superposition of "dead" and "alive" electronic states called here Zombie State (ZS), which can be used in a manner of FCSs but without Grassmann algebra. Instead, for Zombie States, a very simple sign-changing rule is used in the definition of creation and annihilation operators. Then, calculation of electronic structure Hamiltonian matrix elements between two ZSs becomes very simple and a straightforward technique for time propagation of fermionic wave functions can be developed. By analogy with the existing methods based on Canonical Coherent States of Harmonic Oscillator, fermionic wave functions can be propagated using a set of randomly selected Zombie States as a basis. As a proof of principles, the proposed Coupled Zombie States approach is tested on a simple example showing that the technique is exact.

Highlights

In this short paper, a mathematical treatment of second quantization for fermions is proposed, which will be based on the introduction of a simpleminded, Coherent State (CS)like object called here Zombie State (ZS)
This paper introduces a coherent antisymmetrised superposition of “dead” and “alive” electronic states called here Zombie State (ZS), which can be used in a manner of Fermionic Coherent States (FCSs) but without Grassmann algebra
By analogy with the existing methods based on Canonical Coherent States of Harmonic Oscillator, fermionic wave functions can be propagated using a set of randomly selected Zombie States as a basis

Summary

INTRODUCTION

A mathematical treatment of second quantization for fermions is proposed, which will be based on the introduction of a simpleminded, Coherent State (CS)like object called here Zombie State (ZS). Fermionic Coherent States (FCSs) are introduced as |ηm = a(|0m + ηm|1m ), where |1m and |0m are the m-th orbital and its vacuum state, respectively; a is a normalization factor; and ηm is not a number but an element of the Grassmann algebra. Grassmann algebra is needed to ensure correct permutational antisymmetry of multielectron Fermionic Coherent States η = a (|0m + ηm |1m ). Creation and annihilation operators will be defined with the help of a simple sign-changing rule. This rule can be trivially implemented in a computer code and matrix elements of second quantised Hamiltonian can be computed without Wick’s theorem, normally used for standard FCSs.. By analogy with CCS and vMCG, a new class of methods of quantum dynamics of Fermions can potentially be developed with Zombie States

THEORY

WORKING EQUATIONS

PROOF OF PRINCIPLES WITH NUMERICAL EXAMPLES