# Research

# Recent Collaborators

Prof. Fred Manby, University of Bristol, UK

Prof. Jürgen Gauss, Johannes Gutenberg-Universität Mainz, Germany

Prof. Ali Alavi, Max Planck Institute for Solid State Research, Stuttgart, Germany

Prof. Garnet K.-L. Chan, California Institute of Technology, Pasadena, CA, US

Prof. Martin Head-Gordon, University of California, Berkeley, CA, US

Prof. Wenjian Liu, Shandong University, Qingdao, China

Prof. Piotr Piecuch, Michigan State University, MI, US

Prof. Sandeep Sharma, University of Colorado at Boulder, Boulder, CO, US

Prof. Seiichiro L. Ten-no, Kobe University, Kobe, Japan

Prof. Cyrus Umrigar, Cornell University, Ithaca, NY, US

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

## Research Topics

### 7. Mean-Field Density Matrix Decompositions

Lossless property decompositions of HF & KS-DFT into either bond-wise or atomic contributions.

### 6. Contemporary Near-Exact Quantum Chemistry

Assessments of modern near-exact electronic structure theory.

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

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

### 4. 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.

### 2. Local Coupled Cluster Methods

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

### 1. Polarizable Embedding Theory

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

## 7. Mean-Field Density Matrix Decompositions

I've introduced a new and robust decompositions of mean-field Hartree-Fock (HF) and Kohn-Sham density functional theory (KS-DFT) relying on the use of localized molecular orbitals and physically sound charge population protocols. The new lossless property decompositions, which allow for partitioning 1-electron reduced density matrices into either bond-wise or atomic contributions, are intended to be employed in possible applications as an interpretative tool in the rationalization of certain electronic phenomena as well as for exposing and amplifying compositional features in the context of machine-learned quantum chemistry.

The DECODENSE code is open source: https://github.com/januseriksen/decodense

### Relevant Publications:

**7-1****. Eriksen, J. J.***: *Mean-Field Density Matrix Decompositions**.*

**Comparison of bond- and atom-partitioned contributions (in units of Hartree) to the total KS-DFT (TPSSh xc functional) energy of the benzene molecule (Fig. from ref. 7-1).**

## 6. Contemporary Near-Exact Quantum Chemistry

Leading a broad, international consortium of research groups, I've worked on the blind assessment of a wide range of contemporary near-exact electronic structure methods. Specifically, we've reported (in ref. 6-1) on the findings of a blind challenge devoted to determining the frozen-core, full configuration interaction (FCI) ground-state energy of the benzene molecule in a standard correlation-consistent basis set of double-ζ quality. Subsequently, I've been invited to comment on the status of current state-of-the-art approaches in a perspective on the field (ref. 6-2), nearly a century on from the dawn of modern quantum mechanics.

### Relevant Publications:

**6-2****. Eriksen, J. J.**: *The Shape of Full Configuration Interaction to Come**.*

**6-1****. Eriksen, J. J.**; Anderson, T. A.; Deustua, J. E.; Ghanem, K.; Hait, D.; Hoffmann, M. R.; Lee, S.; Levine, D. S.; Magoulas, I.; Shen, J.; Tubman, N. M.; Whaley, K. B.; Xu, E.; Yao, Y.; Zhang, N.; Alavi, A.; Chan, G. K.-L.; Head-Gordon, M.; Liu, W.; Piecuch, P.; Sharma, S.; Ten-no, S. L.; Umrigar, C. J.; Gauss, J.: *The Ground State Electronic Energy of Benzene**.*

**Examples of contemporary near-exact FCI methods (Fig. from ref. 6-2).**

## 5. Many-Body Expanded Full Configuration Interaction

In collaboration with Prof. Jürgen Gauss of the Johannes Gutenberg-Universität Mainz, I've worked on the development of a 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 associated FCI properties. The MBE-FCI method, as implemented in the new PyMBE code, is capable of yielding high-accuracy correlation energies, excitation energies, and (transition) dipole moments for states of both weakly and strongly correlated molecular systems in large basis sets.

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

### Relevant Publications:

**5****-6****. Eriksen, J. J.**; Gauss, J.: *Incremental Treatments of the Full Configuration Interaction Problem.*

**5-5. Eriksen, J. J.**; Gauss, J.: *Ground and Excited State First-Order Properties in Many-Body Expanded Full Configuration Interaction Theory**.*

**5-4. Eriksen, J. J.**; Gauss, J.: *Generalized Many-Body Expanded Full Configuration Interaction Theory*.

**5-3. Eriksen, J. J.**, Gauss, J.: *Many-Body Expanded Full Configuration Interaction. II. Strongly Correlated Regime*.

**5-2. Eriksen, J. J.**, Gauss, J.: *Many-Body Expanded Full Configuration Interaction. I. Weakly Correlated Regime*.

**5-1. 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. 5-2).**

## 4. 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. 4-1 and 4-2), 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. 4-3 and 4-4 - 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:

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

**4-3. 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)

**4-2.** 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*.

**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*.

**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. 4-4). **

## 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).**

## 2. 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. 2-1, and results of the model as well as my other contributions to the DEC family of methods make up parts of refs. 2-2 and 2-3. The DEC-CCSD(T) model currently marks the highest level of complexity among the DEC local correlation methods.

### Relevant Publications:

**2-3.** 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*.

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

**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*.

**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. 2-1).**

## 1. 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. 1-1 and 1-3) and developmental (refs. 1-2 and 1-4) 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. 1-2 and, in particular, ref. 1-3 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. 1-3 was selected as the *Molecular Physics* entry in the *Taylor & Francis Chemistry Top Twenty 2013* selection.

### Relevant Publications:

**1-4. 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*.

**1-3. 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*.

**1-2. 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*.

**1-1. 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*.