# Research

## Research Topics

### 1. Many-Body Expanded Full Configuration Interaction

- Near-exact properties for both weakly and strongly correlated molecular systems in extended basis sets.

### 2. HPC- and GPU-Accelerated Quantum Chemistry

- Efficient implementation of coupled cluster methods on parallel CPU and GPU hardware.

### 3. Coupled Cluster Perturbation Theory

- Theoretical development of Lagrangian-based many-body perturbation theory.

### 4. Local Coupled Cluster Methods

- Implementation and application of the DEC-CCSD(T) computational method.

### 5. Polarizable Embedding Theory

- Polarizable solvent modelling by means of either DFT or many-body methods.

## 1. Many-Body Expanded Full Configuration Interaction

Enabled by an individual postdoctoral fellowship generously granted by the *Alexander von Humboldt Stiftung*, Prof. Jürgen Gauss of the Johannes Gutenberg-Universität Mainz and I have recently initiated work on the development of an alternative and massively parallel route towards the exact solution to the electronic Schrödinger equation in quantum mechanics. The method, denoted the many-body expanded full configuration interaction (MBE-FCI) method, avoids the traditional quest for the complex full configuration interaction (FCI) wave function in favour of the associated FCI energy. The MBE-FCI method, as implemented in the new PyMBE code, is capable of yielding high-accuracy correlation energies for both weakly and strongly correlated molecular systems in large basis sets. The method is currently being extended to the calculation of excitation energies and general first-order properties. Further work on alternative ways of enhancing and accelerating the method even further is currently being pursued in collaboration with Prof. Fred Manby at the University of Bristol as part of an individual postdoctoral fellowship granted by the *Independent Research Fund Denmark*.

The PyMBE code is open source: https://gitlab.com/januseriksen/pymbe

### Relevant Publications:

**1-4. Eriksen, J. J.**, Gauss, J.: *Generalized Many-Body Expanded Full Configuration Interaction Theory.*

**1-3. Eriksen, J. J.**, Gauss, J.: *Many-Body Expanded Full Configuration Interaction. II. Strongly Correlated Regime*.

**1-2. Eriksen, J. J.**, Gauss, J.: *Many-Body Expanded Full Configuration Interaction. I. Weakly Correlated Regime*.

**1-1. Eriksen, J. J.**, Lipparini, F.; Gauss, J.: *Virtual Orbital Many-Body Expansions: A Possible Route Towards the Full Configuration Interaction Limit*.

**Example of the convergence of the MBE-FCI correlation energy from that of a base model (CCSD(T)) towards the exact solution (Fig. from ref. 1-2).**

## 2. HPC- and GPU-Accelerated Quantum Chemistry

Towards the end of my PhD studies under the supervision of Prof Poul Jørgensen at Aarhus University, I got increasingly interested in high-performance computing (HPC) and the modifications required in order to make computer codes run efficiently on modern supercomputers. This work was largely motivated by the fact that our group in Aarhus had been awarded a 3-year (2014-2016) INCITE allocation on the TITAN system @ ORNL, TN, US. Besides internode parallelization by means of the message passing interface (MPI) standard (refs. 2-1 and 2-2), I also took a strong interest in developing quantum chemistry codes for general purpose graphics processing units (GPUs). As examples of GPU-accelerated quantum chemistry, I proposed and demonstrated - in the course of an invited book chapter and a paper, refs. 2-3 and 2-4 - how the non-proprietary OpenMP and OpenACC standards of compiler directives may be used to compactly and efficiently accelerate the rate-determining steps of two of the most routinely applied many-body methods of modern electronic structure theory, namely the RI-MP2 and CCSD(T) models.

### Relevant Publications:

**2-4. Eriksen, J. J.**: *Efficient and Portable Acceleration of Quantum Chemical Many-Body Methods in Mixed Floating Point Precision using OpenACC Compiler Directives*.

**2-3. Eriksen, J. J.**: *Incrementally Accelerating the RI-MP2 Correlated Method of Electronic Structure Theory Using OpenACC Compiler Directives*.

*Parallel Programming with OpenACC*, Ed.: Farber, R., Morgan Kaufmann (2016)

**2-2.** Kjærgaard, T.; Baudin, P.; Bykov, D.; **Eriksen, J. J.**; Ettenhuber, P.; Kristensen, K.; Larkin, J.; Liakh, D.; Pawlowski, F.; Vose, A.; Wang, Y. M.; Jørgensen, P.: *Massively Parallel and Linear-Scaling Algorithm for Second-Order Møller-Plesset Perturbation Theory Applied to the Study of Supramolecular Wires*.

**2-1. Eriksen, J. J.**; Baudin, P.; Ettenhuber, P.; Kristensen, K.; Kjærgaard, T.; Jørgensen, P.: *Linear-Scaling Coupled Cluster with Perturbative Triple Excitations: The Divide-Expand-Consolidate CCSD(T) Model*.

**Total time-to-solution for a CPU-only and a hybrid CPU/GPU implementation (using six K40 NVIDIA GPUs) of the CCSD(T) model for increasingly large alanine (ala) systems (Fig. from ref. 2-4). **

## 3. Coupled Cluster Perturbation Theory

The vast majority of my research during my PhD studies was devoted to the theoretical development of Lagrangian-based CC perturbation theory. In collaboration with Prof. Jürgen Gauss of the Johannes Gutenberg-Universität Mainz and Assist. Prof. Devin A. Matthews of the University of Texas at Austin (now at the Southern Methodist University, Dallas, TX), we advocated (refs. 3-1 and 3-2) and numerically confirmed (refs. 3-3 and 3-4) the existence of a suite of rigorous perturbations series, CC[m_{P}]([m_{Q}]–*n*)* ,* which all expand the difference in energy between any two CC models in orders of the Møller-Plesset fluctuation potential. Besides offering a range of novel perturbational CC models, we were also able to shed new light on potential inconsistencies in the application of more traditional counterparts, e.g., the acclaimed CCSD(T) and CCSDT(Q) models, to open-shell molecular species, cf. refs. 3-5 and 3-6. Finally, the project culminated in two elaborate studies on the general behaviour and convergence of perturbational CC theory in refs. 3-7 and 3-8.

### Relevant Publications:

**3-8. Eriksen, J. J.**; Kristensen, K.; Matthews, D. A.; Jørgensen, P.; Olsen, J.: *Convergence of Coupled Cluster Perturbation Theory*.

**3-7.** Kristensen, K.; **Eriksen, J. J.**; Matthews, D. A.; Olsen, J.; Jørgensen, P.: *A View on Coupled Cluster Perturbation Theory Using a Bivariational Lagrangian Formulation*.

**3-6. Eriksen, J. J.**; Matthews, D. A.; Jørgensen, P.; Gauss, J.: *Assessment of the Accuracy of Coupled Cluster Perturbation Theory for Open-Shell Systems. II. Quadruples Expansions*.

**3-5. Eriksen, J. J.**; Matthews, D. A.; Jørgensen, P.; Gauss, J.: *Assessment of the Accuracy of Coupled Cluster Perturbation Theory for Open-Shell Systems. I. Triples Expansions*.

**3-4. Eriksen, J. J.**; Matthews, D. A.; Jørgensen, P.; Gauss, J.: *Communication: The Performance of Non-Iterative Coupled Cluster Quadruples Models*.

**3-3. Eriksen, J. J.**; Jørgensen, P.; Gauss, J.: *On the Convergence of Perturbative Coupled Cluster Triples Expansions: Error Cancellations in the CCSD(T) Model and the Importance of Amplitude Relaxation*.

**3-2. Eriksen, J. J.**; Jørgensen, P.; Olsen, J.; Gauss, J.: *Equation-of-Motion Coupled Cluster Perturbation Theory Revisited*.

**3-1. Eriksen, J. J.**; Kristensen, K.; Kjærgaard, T.; Jørgensen, P.; Gauss, J.: *A Lagrangian Framework for Deriving Triples and Quadruples Corrections to the CCSD Energy*.

**Schematic representation of the Møller-Plesset (MP) and CC[m _{P}]([m_{Q}]–n) families of perturbation series (Fig. from ref. 3-8).**

## 4. Local Coupled Cluster Methods

Formally, the main topic of my PhD studies was concerned with the development of the CCSD(T) model within the divide-expand-consolidate (DEC) framework for performing local coupled cluster calculations on extended molecular systems. The final massively parallel implementation and proof-of-concept results of the DEC-CCSD(T) model were documented in ref. 4-1, and results of the model as well as my other contributions to the DEC family of methods make up parts of refs. 4-2 and 4-3. The DEC-CCSD(T) model currently marks the highest level of complexity among the DEC local correlation methods.

### Relevant Publications:

**4-3.** Kjærgaard, T.; Baudin, P.; Bykov, D.; **Eriksen, J. J.**; Ettenhuber, P.; Kristensen, K.; Larkin, J.; Liakh, D.; Pawlowski, F.; Vose, A.; Wang, Y. M.; Jørgensen, P.: *Massively Parallel and Linear-Scaling Algorithm for Second-Order Møller-Plesset Perturbation Theory Applied to the Study of Supramolecular Wires*.

**4-2.** Kristensen, K.; Ettenhuber, P.; **Eriksen, J. J.**; Jensen, F.; Jørgensen, P.: *The Same Number of Optimized Parameters Scheme for Determining Intermolecular Interaction Energies*.

**4-1. Eriksen, J. J.**; Baudin, P.; Ettenhuber, P.; Kristensen, K.; Kjærgaard, T.; Jørgensen, P.: *Linear-Scaling Coupled Cluster with Perturbative Triple Excitations: The Divide-Expand-Consolidate CCSD(T) Model*.

**Plot showing CCSD as well as fourth- and fifth-order (T) contributions to the CCSD(T) pair interaction energy as a function of the interatomic pair distance for a cluster of 20 water molecules (Fig. from ref. 4-1).**

## 5. Polarizable Embedding Theory

During my M.Sc. studies at the University of Copenhagen, I was fortunate enough to be offered the chance to lead a number of computational (refs. 5-1 and 5-3) and developmental (refs. 5-2 and 5-4) studies in the area of polarizable solvent modelling. In particular, I took part in an existing collaboration between Stephan P. A. Sauer and Kurt V. Mikkelsen at the University of Copenhagen and Jacob Kongsted and Hans Jørgen Aa. Jensen at the University of Southern Denmark. Through the work in ref. 5-2 and, in particular, ref. 5-3 we were able to highlight the importance and predictive power of polarizable embedding in the description of solvent effects on modest-sized organic molecules. Ref. 5-3 was selected as the *Molecular Physics* entry in the *Taylor & Francis Chemistry Top Twenty 2013* selection.

### Relevant Publications:

**5-4. Eriksen, J. J.**; Solanko, L. M.; Nåbo, L. J.; Wüstner, D.; Sauer, S. P. A..; Kongsted, J.: *The Second-Order Polarization Propagator Approximation (SOPPA) Method Coupled to the Polarizable Continuum Model*.

**5-3. Eriksen, J. J.**; Sauer, S. P. A..; Mikkelsen, K. V.; Christiansen, O.; Jensen, H.-J. Aa.; Kongsted, J.: *Failures of TDDFT in Describing the Lowest Intramolecular Charge-Transfer Excitation in para-Nitroaniline*.

**5-2. Eriksen, J. J.**; Sauer, S. P. A..; Mikkelsen, K. V.; Jensen, H.-J. Aa.; Kongsted, J.: *On the Importance of Excited State Dynamic Response Electron Correlation in Polarizable Embedding Methods*.

**5-1. Eriksen, J. J.**; Olsen, J. M.; Aidas, K.; Ågren, H.; Mikkelsen, K. V.; Kongsted, J.: *Computational Protocols for Prediction of Solute NMR Relative Chemical Shifts. A Case Study of L-Tryptophan in Aqueous Solution*.

# Recent Collaborators

- Prof. Fred Manby, University of Bristol, UK
- Prof. Jürgen Gauss, Johannes Gutenberg-Universität Mainz, Germany
- Assist. Prof. Filippo Lipparini, Università di Pisa, Italy
- Assist. Prof. Devin A. Matthews, Southern Methodist University, Dallas, TX, US
- Prof. Poul Jørgensen, Aarhus University, Denmark
- Prof. Jeppe Olsen, Aarhus University, Denmark