Highly Accurate Molecular Properties using variational Quantum Electrodynamics

Project Information

HAMP-vQED

Grant agreement ID: 101019907

DOI

10.3030/101019907

EC signature date 3 May 2021

Start date 1 October 2021

End date 30 September 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 929 658,00

EU contribution

€ 1 929 658,00

Coordinated by

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 2 - HAMP-vQED (Highly Accurate Molecular Properties using variational Quantum Electrodynamics)

Reporting period: 2023-04-01 to 2024-09-30

Summary of the context and overall objectives of the project

Does chemistry need more physics ?

The building blocks of chemistry are organized in the periodic table, which requires quantum mechanics for its understanding. In the 1980s it was realized that in order to correctly describe the heavy elements in the lower part of the periodic table, the special theory of relativity had to be invoked. The HAMP-vQED projet will investigate the possible role of quantum electrodynamics (QED) in chemistry. It will provide a protocol for highly accurate calculations of molecular properties, with particular attention to properties that probe the electron density in the close vicinity of atomic nuclei, where the QED-effects associated with the Lamb shift are created. The HAMP-vQED adheres to the general framework of quantum chemistry by seeking a variational (non-perturbative) approach using local (Gaussian) basis functions.

Is the vacuum really empty ?

An ideal sitatution for accurate calculations is a molecule alone in space at 0K. However, is the vacuum really empty ? It has been shown that placing a molecule in an otherwise empty cavity will change its reactivity. This is explained by the coupling of the molecule to the zero-point vibrations of the quantized electromagnetic field.
In the HAMP-vQED we are particularly interested in the effects leading to the Lamb shift, a splitting between the 2S1/2 and 2P1/2-levels of one-electron atoms, not predicted by the Dirac equation, but observed for the first time in 1947 by Lamb and Retherford.

* Vacuum polarization: A charge in space is surrounded by virtual electron-positron pairs. This will contribute to the observed charge.
* Electron self-energy: A charge drags along its electromagnetic field. This will contribute to the observed mass.

The splitting is a mere 4 meV, but for hydrogen-like uranium the splitting has grown to an impressive 76 eV. It is therefore legitimate to ask if QED-effects could play a role in the chemistry of heavy elements.

Project objectives

1. Set new standards for correlated relativistic molecular calculations, with particular focus on properties probing nuclear regions
2. Develop a variational approach to QED rather than the usual perturbative one (QED without diagrams)

The most challenging part of the project is to devise ways of handling the divergences of QED using the computational framework of quantum chemistry.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

1. Beyond the no-pair approximation
================================

1.1 Effective QED potentials
-----------------------------------
We have reported the implementation of three such potentials in the DIRAC program package [https://doi.org/10.1063/5.0116140]: the Uehling potential for vacuum polarization (VP) as well as two effective potentials for electron self-energy.

1.2 Towards variational QED
-----------------------------------
At the lowest level of bound-state QED in the S-matrix formalism one finds three contributions: i) Vacuum polarization ii) Electron self-energy and iii) Single-photon exchange [relativistic two-electron interaction]

1.2.1. Vacuum polarization (VP)
~~~~~~~~~~~~~~~~~~~~~~~~~~~
In the atomic case, upon expansion of the VP density in orders of the nuclear charge, the linear term is divergent and the physical part, generating the Uehling potential, is only obtained after charge renormalization. For the non-linear part, we have developed a robust numerical machinery for the calculation of the non-linear part of the VP density [https://doi.org/10.1103/PhysRevA.108.012808] in a finite Gaussian basis. For the linear part, we are in the process of devising schemes for regularization and renormalization in a finite basis. We have developed a way to introduce a sharp momentum cutoff in a finite basis and are currently exploring its effect. We are also exploring the Pauli-Villars regularization scheme based on the use of auxiliary masses.

1.2.2 Electron self-energy
~~~~~~~~~~~~~~~~~~~~~~~
We are investigating the partial wave renormalization scheme for calculating the electron self-energy in atomic systems, using finite Gaussian basis sets.

1.2.3 Relativistic two-electron interaction
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Work is in progress, but not completed, for the implementation of the Breit two-electron term at the SCF and correlated levels. We have implemented the Gaunt term (part of the Breit term) for the calculation of magnetic properties at the SCF level.

2.Coupled-cluster response theory
==============================

We have developed a general-order coupled-cluster code generator, written in python and named tenpi. Using diagrammatic methods the basic equations are formulated and then both flop count and memory is optimized by defining suitable intermediates. The generator produces several output code types, including Fortran code for modern distributed memory tensor libraries designed to run on supercomputers, but also a python code which allows for rapid prototyping and testing of methods. The generated Fortran code has been included into the DIRAC program package and enabled it for the first time to run CCSDT and CCSDTQ calculations on multiple computing nodes in parallel.
Our first target property is the electric field gradient (EFG) at nuclear positions in a molecule, which, combined with experiment, allows to extract the value of nuclear electric quadrupole moments. In preparation for more accurate calculation, we have carried out a detailed study at the DFT level of the chemical information that can be extracted from knowledge of the EFG.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

1. Beyond the no-pair approximation
================================

1.1 Effective QED potentials
-----------------------------------
We are planning to implement the effective self-energy potential of Shabaev et al. [https://doi.org/10.1103/PhysRevA.88.012513]. All effective QED-potentials are to be tested for core properties.

1.2 Towards variational QED
-----------------------------------
In the terminology of Hartree-Fock theory, vacuum polarization and self-energy correspond to direct and exchange contributions, respectively.
The most challenging part of the project is to devise techniques for regularisation and renormalization in coordinate space for use with finite basis sets. We will pursue our efforts in this direction.
We will investigate whether the by now well-established algorithms for converting delocal Hartree–Fock exchange into a local, so-called optimum effective potential can be used to convert the delocal self-energy into a local effective QED-potential.

1.2.3 Relativistic two-electron interaction
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We will pursue the implementation of the Breit two-electron term, including frequency-dependence.

2.Coupled-cluster response theory
==============================

We have learned from our high-level coupled cluster calculations so far that it is crucial to reduce storage of coupled-cluster amplitudes. We aim to explore index permutation antisymmetry as well as block sparsity from spatial symmetry to achieve this.
We will progressively implement expectation values and response functions at the relativistic coupled-cluster level. A particular challenge is the formulation of magnetic properties within the no-pair approximation since the diamagnetic contributions is built from negative-energy orbitals.
These high-level methods will be combined with the treatment of QED-effects to provide highly accurate of molecular core properties.

The periodic table (1871)

The target QED-effects arise from the polarizable vacuum.

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