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## Stochastic Wavefunction Methods in Quantum Chemistry, Electronic Structure Theory and Condensed Matter Physics

#### CECAM-HQ-EPFL, Lausanne, Switzerland

#### Organisers

Electronic structure theory faces many exciting new challenges in the coming years. These are dominated by the challenge of solving the many-body problem to high accuracy, the need to achieve this consistently for large systems, and the requirement to adapt software to high-performance computing in the petascale era. Monte Carlo methods are immediately suited to meet these problems, and a wealth of stochastic algorithms for the numerical solutions arising in quantum chemistry and electronic structure have arisen in recent years.

Full configuration interaction quantum Monte Carlo [1] is an efficient algorithm to perform a ground-state diagonalisation of the many-body Hamiltonian in the Slater determinant basis, designed for use in molecular systems. This was achieved by using a projector method, and the imaginary-time Schrodinger equation, in common with other stochastic algorithms such as diffusion Monte Carlo, auxilliary-field quantum Monte Carlo or Greens’ function Monte Carlo. The method was extended a coupled of years after by the development of the initiator algorithm (i-FCIQMC) which allowed for a substantial alleviation of the sign problem and the treatment of significantly larger spaces [2]. Early applications included atoms and molecules [3, 4], but now the range of realistic systems also extends to the solid state [5]. The largest systems attempted to date are those where the integrals can be computed analytically, namely the uniform electron gas [6] and Hubbard model [7]. There is also interest in using these methods to treat infinite nuclear matter.

There followed the rapid development a range of varyingly related novel stochastic electronic structure approaches. Very broadly, there were two categories of these. Algorithmic developments (where FCI was still being solved) and methodological developments (where the objective changed).

Algorithmic developments include semi-stochastic QMC (SQMC), which treats part of the

Slater determinant space deterministically, offering a substantial reduction in the stochastic error [8]; and partial-node configuration interaction QMC [9], which attempts to remove the sign problem by restricting the signs of the determinants to those of a trial wave function. In a closely-related configuration interaction Monte Carlo, a guiding function was used for similar purposes from coupled-cluster calculations [10].

New methodologies have also been developed around the same core algorithm as FCIQMC including a stochastic coupled-cluster theory (CCMC) [11]; a model-space quantum Monte Carlo (MSQMC) method able to solve for target excited states [12]; and density-matrix quantum Monte Carlo (DMQMC), which samples the N-particle density matrix to obtain finite-temperature properties [13].

Success in developing stochastic methodologies has not been limited to FCIQMC and related algorithms; this workshop wishes to make links between this method and other emerging ideas.

Promising work has been made with stochastic implementations of MP2 in recent years [14–16]. The MP2 equations are sufficiently simple that they can be sampled directly via Monte Carlo, but such attempts do not achieve speed-ups that translate into widespread improvement across a range of systems. Numerical transformations for the denominator, and good biasing functions, have been particularly important [14–16]. It is encouraging to see that progress is also being made at treating F12 methods stochastically [17].

It is important to observe that useful stochastic approaches to the many-electron problem can often be based on quantum chemical algorithms methods that are several generations of technology out of date. Resource needs to be targeted at what is already known from the pre-existing community. Here, in particular, there have been significant recent developments in reducing the scaling of the random phase approximation (typically, with a method by which screening can be achieved). This is a method of particular interest because its suitability for solid state systems [18] and its relationship with density functional theory and coupled cluster theory [19, 20].

It therefore seems an appropriate time to gather the community of emerging stochastic quantum chemists in an interdisciplinary environment to address the following challenges:

(1) The proper understanding of the sign problem, systematic undersampling and stochastic error;

(2) The need to keep pace with algorithmic development that lowers the cost/scaling of conventional techniques;

(3) To provoke information and technology transfer between different types of quantum Monte Carlo;

(4) To direct efforts towards new applications of widespread interest that are well-suited to Monte Carlo sampling;

(5) To encourage development of deterministic techniques by stochastic benchmarking, and set challenges for stochastic techniques by deterministic analysis.

To attempt to achieve progress, we envisage this workshop to combine users and developers of existing stochastic quantum chemistry techniques, those working in the surrounding electronic structure community on low-scaling algorithms and cutting-edge applications.

In summary, we wish to draw together the new community working on stochastic electronic structure with a particular emphasis on those working on Fock space methods with annihilation. Through engagement with other electronic structure theorists, we will address specific challenges that need to be met in the coming years in order to establish stochastic methods as useful to developers in a wide range of communities.

## References

**United Kingdom**

Matthew Foulkes (Imperial College London) - Organiser

**United States**

James J. Shepherd (Massachusetts Institute of Technology) - Organiser