Research interests

My research work focuses on the numerical and theoretical analysis of numerical models and methods in quantum chemistry, whether they come from solid state physics or molecular chemistry. More precisely, they aim to estimate the errors that have several possible origins

  1. modeling errors: the equations which govern quantum chemistry are known - the Schrödinger equation is a high-dimensional linear PDE - which can only be solved for systems reduced to a few electrons. The models (Hartree-Fock or Kohn-Sham) reduce the problem to solving a nonlinear system of PDEs in 3D.

  2. discretization errors: for a given model, this error depends on the numerical scheme considered (plane waves, finite elements or finite differences).

  3. algorithmic errors: at a fixed model and scheme, the chosen algorithm introduces an error which may be difficult to assess. The nonlinear PDEs in quantum chemistry are solved by fixed point methods. The underlying problem being in the vast majority of cases non-convex, the convergence with respect to a numerical tolerance is unclear.

Tensor Networks

Tensor Networks (TN) have seen rapid and emerging interests. They aim to give a sparse representation of high-dimensional functions. The first instance of TN was Matrix Product States (MPS) or Tensor Trains that were introduced in the late 90s/early 2000s in physics as DMRG (Density Matrix Renormalization Group) to find the ground-state of statistical models (one-dimensional Hubbard or Heisenberg models). MPS has been used for the NN-body electronic Schrödinger problem where the question of the optimal topology (or best ordering) is not clear.

In the contribution, we investigate the behaviour of the singular values of reshapes of a particular class of functions (Slater determinants). The decay of these singular values drives the approximability of the function by an MPS. For this class of functions, if we denote by (σj)(\sigma_j) the singular values of the reshaped tensor Ψμ1μL/2μL/2+1μL\Psi_{\mu_1 \dots \mu_{L/2}}^{\mu_{L/2+1}\dots \mu_L}, then we show that there are 2N2^N positive singular values, where NN is the number of particles and they satisfy

σ1σ2N=σ2σ2N1=σ3σ2N2==Const. \sigma_1 \sigma_{2^N} = \sigma_2 \sigma_{2^N-1} = \sigma_3 \sigma_{2^N-2} = \cdots = \mathrm{Const}.

The constant is moreover explicit and depends on the ordering of the modes of the tensor. This provides a natural criterion to optimze a TT approximation of the state.

 Singular values of the reshaped tensor for different ordering schemes
Singular values of the reshaped tensor for different ordering schemes

Publication

Solid state physics

In solid-state physics, the periodic boundary conditions motivate a plane-wave discretisation. However because of the Coulomb singularities at the nuclei, the eigenfunctions of the Hamiltonian Δ+V-\Delta + V have cusps which impedes the convergence. Pseudopotential methods are widely used, where the Coulomb singularities are smoothened. In the PAW method (Projector Augmented-Wave), introduced by Blöchl, the pseudopotential is introduced using an invertible operator acting locally around the nuclei. The PAW equations involve infinitely many terms that are truncated in the numerical treatment which causes a modelling error. This modelling error has been analysed for the one-dimensional model d2dx2Z0kZδkZakZδa+k-\frac{\mathrm{d}^2}{\mathrm{d}x^2} - Z_0 \sum_{k \in \mathbb{Z}} \delta_k - Z_a\sum_{k \in \mathbb{Z}} \delta_{a+k} for which explicit formulas of the eigenpairs are available.

Another approach consists in introducing an invertible operator TT acting locally around the nuclei. This operator is defined such that it maps smooth functions to functions with cusps satisfying the Kato cusp condition. This operator acts as a preconditonner for a plane-wave discretisation. This method is called the Variational Projector Augmented-Wave and has been analysed for one-dimensional and three-dimensional Hamiltonians.

 Comparison between a direct discretization and the VPAW method for $-\Delta - \frac{1}{|x-R|}-\frac{1}{|x+R|}$ with periodic boundary conditions
Comparison between a direct discretization and the VPAW method for $-\Delta - \frac{1}{|x-R|}-\frac{1}{|x+R|}$ with periodic boundary conditions

Publications

Anderson acceleration

Anderson acceleration, also known as DIIS (Direct Inversion of the Iterative Space) in quantum chemistry, aims to accelerate the convergence of fixed-point iteration xk+1=g(xk)x_{k+1} = g(x_k). Such fixed-point problems are ubiquitous, and in quantum chemistry come from the Hartree-Fock or Kohn-Sham equations. DIIS is an extrapolation method where the next iterate is built using the knowledge of previous iterates. The DIIS sequence is given by

(ci(k))0im=arg minci=1i=0mcif(xki)2andxk+1=g(i=0mci(k)xki), (c^{(k)}_i)_{0 \leq i \leq m} = \argmin_{\sum c_i = 1} \left\|\sum\limits_{i=0}^m c_i f(x_{k-i})\right\|_2 \quad \text{and} \quad x_{k+1} = g\left(\sum_{i=0}^m c^{(k)}_i x_{k-i}\right),

where ff is an error function that locally vanishes only at the solution of the fixed-point problem. A usual choice is f(x)=xg(x)f(x) = x-g(x).

An important parameter in the efficiency of the method is the width mm of the history/number of iterates kept. Two variants are considered in our publication:

For both variants, we prove local linear convergence and superlinear convergence by tuning the DIIS numerical parameters.

 Comparison between the DIIS variants with fixed depth (FD-DIIS), with restarts (R-DIIS) and adaptive depth (AD-DIIS)
Comparison between the DIIS variants with fixed depth (FD-DIIS), with restarts (R-DIIS) and adaptive depth (AD-DIIS)

Publication

Linear response in quantum chemistry

Properties of a molecule in its ground-state perturbed by a time-dependent potential can be approximated in the linear regime (assuming that the perturbation is small) using the Kubo formula. To be more precise, if ψ0\psi_0 is the ground-state of H=Δ+VH = -\Delta + V an operator acting on L2(Rd)L^2(\mathbb{R}^d), for some observable VOV_\mathcal{O} and ψ(t)\psi(t) solution to

itψ=Hψ+εf(t)VPψ,ψ(0)=ψ0, i\partial_{t} \psi = H \psi + \varepsilon f(t) V_{\mathcal P} \psi , \quad \psi(0) = \psi_{0},

the linear response function KK defined by

ψ(t),VOψ(t)=ψ0,VOψ0+ε(Kf)(t)+O(ε2), \langle \psi(t), V_{\mathcal O} \psi(t) \rangle = \Big\langle \psi_{0}, V_{\mathcal O} \psi_{0}\Big\rangle + \varepsilon (K \ast f)(t) + \mathcal{O}(\varepsilon^2),

is given by the Kubo formula

K(τ)=iθ(τ)VOψ0,ei(HE0)τVPψ0+c.c.. K(\tau) = -i \theta(\tau) \Big\langle V_{\mathcal O} \psi_{0}, e^{-i (H-E_{0}) \tau} V_{\mathcal P} \psi_{0}\Big\rangle + {\rm c.c.}.

The Fourier transform of KK is then

Kundefined(ω)=limη0+ψ0,VO(ω+iη(HE0))1VPψ0ψ0,VP(ω+iη+(HE0))1VOψ0. \widehat K(\omega) = \lim_{\eta \to 0^{+}} \Big\langle \psi_{0}, V_{\mathcal O} \Big(\omega +i\eta - (H-E_0)\Big)^{-1} V_{\mathcal P} \psi_{0}\Big\rangle - \Big\langle \psi_{0},V_{\mathcal P} \Big(\omega +i\eta + (H-E_0)\Big)^{-1} V_{\mathcal O} \psi_{0}\Big\rangle.

The above formula is in practice approximated by taking a positive η\eta and by truncating the space to a finite region (L,L)d(-L,L)^d. This necessarily yields a singular approximation of the linear response function Kundefined\widehat{K}. In the paper, the regularity of the response function Kundefined\widehat{K} is determined, depending on the decay of the potential VV. Moreover, the relationship between η\eta and LL guaranteeing the convergence of the approximation is shown, supported by numerical experiments.

Publication