Licence / Master Science de la matière  Stage 2019-2020
École Normale Supérieure de Lyon  Antoine MARIE
Université Claude Bernard Lyon I  M1 Chimie

Pertubation theories in the complex plane

Abstract : Un résumé de 10 lignes environ du contenu du rapport, permettant de situer le sujet et les résultats principaux du stage

Keywords : perturbation theory, Hartree-Fock, spherium, exceptional points

Internship supervisor: Pierre-François LOOS

loos@irsamc.ups-tlse.fr/ tél. (+33) 5 61 55 73 39

Laboratoire de Chimie et Physique Quantiques 118, route de Narbonne 31062 Toulouse - France

https://www.irsamc.ups-tlse.fr/

July 6, 2020

PT -symmetric electronic structure theory  Antoine MARIE

Acknowledgments Contents

1 Introduction  3
2 Perturbation theory  3
3 Historical overview  4

PT -symmetric electronic structure theory  Antoine MARIE

1 Introduction

It has always been of great importance for theoretical chemists to better understand excited states and their properties because processes involving excited states are ubiquitous in nature (physics, chemistry and biology). One of the major challenges is to accurately compute excited states energies. Plenty of methods have been developed to this aim and each of them have its own qualities but also its own flaws. The fact that none of all those methods is succesfull for every molecule in every geometry encourage chemists to continue the development of new methodologies to get accurate energies and to try to understand deeply why each methods fails or not with each molecule. All those methods rely on the notion of quantised energy levels of Hermitian quantum mechanics. In quantum chemistry, the ordering of the energy levels represents the different electronic states of a molecule, the lowest being the ground state while the higher ones are the so-called excited states. And we need methods to get accurately how those states are ordered. Within this quantised paradigm, electronic states look completely disconnected from one another. However, one can gain a different perspective on quantisation if one extends quantum chemistry into the complex domain. In a non-Hermitian complex picture, the energy levels are sheets of a more complicated topological manifold called Riemann surface, and they are smooth and continuous analytic continuation of one another. In other words, our view of the quantised nature of conventional Hermitian quantum mechanics arises only from our limited perception of the more complex and profound structure of its non-Hermitian variant. Therefore, by analytically continuing the energy E(λ) in the complex domain (where λ is a coupling parameter), one can smoothly connect the ground and excited states of a molecule. This connection is possible because, by extending real numbers to the complex domain, one loses the ordering property of real numbers. Hence, one can interchange electronic states away from the real axis, as the concept of ground and excited states has been lost. Amazingly, this smooth and continuous transition from one state to another has been recently realized experimentally in physical settings such as electronics, microwaves, mechanics, acoustics, atomic systems and optics. [?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?] Exceptional points (EPs) [?, ?, ?, ?] are non-Hermitian analogs of conical intersections (CIs) [?] where two states become exactly degenerate. CIs are ubiquitous in non-adiabatic processes and play a key role in photo-chemical mechanisms. In the case of auto-ionizing resonances, EPs have a role in deactivation processes similar to CIs in the decay of bound excited states. Although Hermitian and non-Hermitian Hamiltonians are closely related, the behavior of their eigenvalues near degeneracies is starkly different. For example, by encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. Additionally, the wave function picks up a geometric phase (also known as Berry phase [?]) and four loops are required to recover the starting wave function. In contrast, encircling Hermitian degeneracies at CIs introduces only a geometric phase while leaving the states unchanged. More dramatically, whilst eigenvectors remain orthogonal at CIs, at non-Hermitian EPs the eigenvectors themselves become equivalent, resulting in a self-orthogonal state. [?] More importantly here, although EPs usually lie off the real axis, these singular points are intimately related to the convergence properties of perturbative methods and avoided crossing on the real axis are indicative of singularities in the complex plane. [?, ?]

2 Perturbation theory

Within perturbation theory, the Schrödinger equation is usually rewritten as

H(λ)Ψ = (H^{(0)} + λV)Ψ(λ) = E(λ)Ψ,  (1)

with

V = H − H^{(0)}  (2)

The energy can then be written as a power series of λ

E(λ) = ∑_{k=0}^{∞} λ^{k}E^{(k)}  (3)

where λ is a coupling parameter set equal to 1 at the end of the calculation. However it is not guaranteed that the series E(λ) has a radius of convergence |λ_{0}| < 1. It means that the series is divergent for the physical system at λ = 1. One can prove that |λ_{0}| can be obtained by extending λ in the complex plane and looking for the singularities of E(λ). Theorem: The radius of convergence of the expansion of a function is equal to the distance of the closest singularities in the complex plane to the origin. These singularities are nothing but EPs at λ_{0} and λ_{0}^{∗}. Indeed, for a Hermitian Hamiltonian the singularities occur in complex conjugate pairs with non-zero imaginary parts. The discovery of a partitioning of the Hamiltonian that allowed chemists to get access to a part of the correlation energy using perturbation theory has been a major step in the development of post-Hartree-Fock methods. This case of the Rayleigh-Schrödinger perturbation theory is called the Möller-Plesset perturbation theory. In the MPPT the unperturbed Hamiltonian is the sum of the mono-electronic Fock operator.

H_{0} =  (4)

In Hartree-Fock theory you approximate the exact wave function as a Slater-determinant (which is an anti-symmetric combination of mono-electronic orbitals) and those wave functions are eigenvectors of the Fock operators. In the perturbation theory the energy is a power series of λ and the physical energy is obtained by taking λ equal to 1. We will refer to there energy up to the n-th order as the MPn energy. The MP1 energy is the Hartree-Fock energy and the correlation energy is getting recovered from the MP2 level.

E =  (5)

But a priori there is no reason that this power series is always convergent for λ=1. In fact, it is known that when the Hartree-Fock wave function is a bad approximation of the exact wave function, for example for multi-reference states, the Möller-Plesset will give bad results. A smart way to investigate the convergence properties of the MP series is to transform the coupling parameter λ into a complex variable. By doing this the Hamiltonian and the energy become functions of this variable. So by searching the singularities of the function E(λ) we can get information on the convergence properties of the MPPT. Those singularities of the energy are exactly the exceptional points connecting the electronic states mentioned in the introduction. The direct computation of the terms of the series is relatively easy up to the 4th order and the 5th and 6th order can be obtained at high cost. But to understand deeply the behavior of the MP series and how it is connected to the singularities, we need to have access to high order terms of the series. For small systems we can have access to the whole series using Full Configuration Interaction. If you diagonalize the Hamiltonian H(λ) in the FCI basis you get the exact energies (in this finite basis set) and expanding in λ allow you to get the Möller-Plesset perturbation series at every order.

3 Historical overview

First the chemists hoped that each time you compute a term of higher order of the perturbation expansion the energy would be more accurate. If this was true it would give a direct method to the exact energy in a finite basis. But rapidly some strange behaviors of the series have been exhibited. In the late 80’s Gill et al. have reported deceptive and slow convergences in stretch systems. In the figure below we can see that the restricted Möller-Plesset series is convergent but oscillating which is not convenient if you are able to compute only few terms (for example RMP5 is worse than RMP4 in this case).

The location of singularities in the complex plane is already known as important for theoretical chemists within the framework of an other method: perturbation theory. One way to improve the accuracy of the energies obtained with the Hartree-Fock method is to use the Rayleigh-Schrödinger Perturbation Theory formalism with the HF Hamiltonian as the unperturbed Hamiltonian, this is known as the Möller-Plesset Perturbation Theory (MPPT). The MPPT has been widely used and it was first thought that increasing the order of the perturbation will increase the accuracy of the energy and that this is the ability to compute the terms of superior order that will limit the accuracy. Using Full Configuration Interaction (FCI) one can have access to high order terms of the perturbation expansion for small systems and it has enabled chemists to prove that the previous assumptions is false. Gill et al. have shown that the MP series in the Unrestricted Hartree-Fock (UHF) framework can exhibits deceptive or erratic convergence. And even more surprising some cases of divergence have also been highlighted by Olsen et al. in systems considered as well understood (Ne, HF and H20). By transforming the "order parameter" λ in Eq 1 in a complex variable thus the Hamiltonian and the energy become a function of this variable and it has been proved that the radius od convergence of the perturbation series is linked to the nearest singularity of the origin of E(λ).

H(λ) = H_{0} + λ(V − H_{0})  (6)

One can proves that if the Hamiltonian is Hermitian then the singularities are conjugate pair of square root branch points with non-imaginary parts, these are the EPs mentionned above. Analysing the location of EP through the formalism of pertubation theory