Physical Chemistry and Theoretical Chemistry (Council of Physical Sciences)

80 %

Ortelius classification: Quantum chemistry

Mathematics (Council of Physical Sciences)

20 %

Ortelius classification: Applied mathematics

Panel

Chemistry 1

Department or equivalent

Theoretical Solid State Physics Department (Wigner Research Centre for Physics of the Hungarian Academy of Sciences)

Participants

Nagy, Péter Sólyom, Jeno Szalay, Szilárd Veis, Libor

Starting date

2013-09-01

Closing date

2017-08-31

Funding (in million HUF)

28.792

FTE (full time equivalent)

3.44

state

closed project

Summary in Hungarian

A kutatás összefoglalója, célkitűzései szakemberek számára Itt írja le a kutatás fő célkitűzéseit a témában jártas szakember számára. The computation of the electronic structure is of utmost importance for the task of molecular engineering in modern chemistry and material science. In this context, the accurate computation of the electron correlation is a fundamental and extremely difficult problem. In contrast to the tremendous progress made in calculating weakly correlated systems by Density Functional Theory (DFT) for extended systems or Coupled Cluster Methods for highly accurate calculations, there are two major types of systems for which current quantum chemical methods have deficiencies: (1) Open-shell systems with a large number of unpaired electrons, as they occur in multiple transition metal complexes or in molecular magnets; (2) Extended or periodic systems without a band gap, where the limit of the applicability of the available size consistent methods is reached. The aim of this proposal is to develop a general tensor network state (TNS) based algorithm that can be applied efficiently to these open problems of quantum chemistry. Realization of such an algorithm relies on carrying out a variety of complex tasks. Several new formal methods and methodological concepts of tensor decompositions will have to be designed to comply with the specific, nonlocal nature of the Hamiltonian, and to this end, the applicants will join their rather complementary expertise regarding the powerful DMRG method and similar recent developments from physics, mathematics and information technology. To arrive at an efficient implementation of the quantum chemistry TNS algorithm, our contributions will be implemented and tested based on existing program structures of the QC-DMRG-Budapest and the TTNS-Vienna codes.

Mi a kutatás alapkérdése? Ebben a részben írja le röviden, hogy mi a kutatás segítségével megválaszolni kívánt probléma, mi a kutatás kiinduló hipotézise, milyen kérdéseket válaszolnak meg a kísérletek. For the approximation of the wave function of an electronic system, any method chosen will have to compromise between the demanded accuracy on the one hand and the high computational complexity of the task on the other. While density functional theory (DFT) and Coupled Cluster approaches are in this sense standard methods for the quantitative study of large, resp., the qualitative study of small weakly correlated systems, there is so far no method-of-choice solution to finding a sufficiently accurate, data-sparse representation of the exact many-body wave function if the electrons are strongly correlated, as, for instance, in high spin open-shell systems as transition metal systems. Due to the present many-electron interactions, strongly correlated problems cannot be sufficiently described by small perturbations of a single Slater determinant. For the treatment of other many-particle systems, e.g., spin systems, alternative representations have been proposed, resulting in the development of so-called matrix product states (MPS). These represent a system of N components or “sites” (corresponding, e.g., to molecular orbitals) by forming products of N matrices, each belonging to one component of the system. The aim of the present project is to develop and implement an efficient quantum chemistry algorithm based on tree tensor network states (QC-TNS), in particular enabling the treatment of problems in quantum chemistry that are intractable by standard techniques as DFT or Coupled Cluster.

Mi a kutatás jelentősége? Röviden írja le, milyen új perspektívát nyitnak az alapkutatásban az elért eredmények, milyen társadalmi hasznosíthatóságnak teremtik meg a tudományos alapját. Mutassa be, hogy a megpályázott kutatási területen lévő hazai és a nemzetközi versenytársaihoz képest melyek az egyediségei és erősségei a pályázatának! Even though Density Functional Theory (DFT) methods became dominant in the field of electronic structure calculations, there are some special areas where their accuracy is not satisfactory. The applicability of highly accurate multi-reference calculations for such problems is limited to relatively small systems due to their high computational cost. Therefore, computationally less demanding new methods offering ways to accurately compute the electronic correlation are of high importance. In this respect, the QC-TNS, development of which is subject to the current proposal, combines a number of favorable features that suggest it might represent a novel, flexible approach in quantum chemistry: The more general concept of data-sparsity inherent in the TNS representation allows for the efficient representation of a much bigger class of wave functions than accessible by state-of-the-art methods. The desired accuracy may be adjusted, so that the ansatz in principle permeates the whole full-CI space. Additionally, the ansatz is size-consistent by construction and capable of calculating excited states and degenerated eigenfunctions[Legeza-2003b,Dorand-2007]; additional symmetries as particle number and spin symmetries and even non-Abelian symmetries are factored out explicitly [McCulloch-2002,Legeza-2003b,Toth-2008]. The approach might therefore provide a viable alternative in the computation of standard situations, e.g., it may be used to obtain high-precision ab initio reference data for DFT calculations. Further, we are also confident that it offers a solution for the computation of some up to now problematic systems, such as high spin open-shell transition metal complexes [Marti-2008, Marti-2010a, Reiher-2009, Kurashige-2009], where due to heavy elements relativistic effects must be taken into account.

A kutatás összefoglalója, célkitűzései laikusok számára Ebben a fejezetben írja le a kutatás fő célkitűzéseit alapműveltséggel rendelkező laikusok számára. Ez az összefoglaló a döntéshozók, a média illetve az adófizetők tájékoztatása szempontjából különösen fontos az NKFI számára. The QC-TNS will be built using the experiences and also partially on the structures of two codes, the QC-DMRG-Budapest, designed for MPS computations, and the TTNS-Vienna code, a prototype for tree tensor network computations. The QC-DMRG-Budapest code has first been applied in quantum chemistry application since 2001, and has due to its efficiency recently become the basic tool to carry out large scale DMRG computations in the group of Prof. M. Reiher at ETH Zürich. Among other concepts that ensure fast and stable convergence and optimized use of computational resources, the program features dynamic rank selection (DBSS), optimization of initialization steps (CI-DEAS), partial summation of auxiliary operators, efficient treatment of excited states, treatment of orbital spatial symmetries as well as Abelian and non-Abelian symmetries. The algorithm may be used for the calculation of spin density distributions for molecules that require very large active spaces. Optimization of network structures based on quantum information entropy and orbital of orbital bases are also taken into account to a certain level. These results confirm that the entropy optimized QC-DMRG can provide adequate solutions for difficult problems in transition-metal chemistry, and in fact, it can be used as a ’black box’ method. The TTNS-Vienna code has been developed quite recently and provides a very flexible class structure to handle the more general TNS topology. At present stage it can, however, only be applied to relatively small systems, since many important features related to the QC applications are not implemented yet.

Summary

Summary of the research and its aims for experts Describe the major aims of the research for experts. The computation of the electronic structure is of utmost importance for the task of molecular engineering in modern chemistry and material science. In this context, the accurate computation of the electron correlation is a fundamental and extremely difficult problem. In contrast to the tremendous progress made in calculating weakly correlated systems by Density Functional Theory (DFT) for extended systems or Coupled Cluster Methods for highly accurate calculations, there are two major types of systems for which current quantum chemical methods have deficiencies: (1) Open-shell systems with a large number of unpaired electrons, as they occur in multiple transition metal complexes or in molecular magnets; (2) Extended or periodic systems without a band gap, where the limit of the applicability of the available size consistent methods is reached. The aim of this proposal is to develop a general tensor network state (TNS) based algorithm that can be applied efficiently to these open problems of quantum chemistry. Realization of such an algorithm relies on carrying out a variety of complex tasks. Several new formal methods and methodological concepts of tensor decompositions will have to be designed to comply with the specific, nonlocal nature of the Hamiltonian, and to this end, the applicants will join their rather complementary expertise regarding the powerful DMRG method and similar recent developments from physics, mathematics and information technology. To arrive at an efficient implementation of the quantum chemistry TNS algorithm, our contributions will be implemented and tested based on existing program structures of the QC-DMRG-Budapest and the TTNS-Vienna codes.

What is the major research question? Describe here briefly the problem to be solved by the research, the starting hypothesis, and the questions addressed by the experiments. For the approximation of the wave function of an electronic system, any method chosen will have to compromise between the demanded accuracy on the one hand and the high computational complexity of the task on the other. While density functional theory (DFT) and Coupled Cluster approaches are in this sense standard methods for the quantitative study of large, resp., the qualitative study of small weakly correlated systems, there is so far no method-of-choice solution to finding a sufficiently accurate, data-sparse representation of the exact many-body wave function if the electrons are strongly correlated, as, for instance, in high spin open-shell systems as transition metal systems. Due to the present many-electron interactions, strongly correlated problems cannot be sufficiently described by small perturbations of a single Slater determinant. For the treatment of other many-particle systems, e.g., spin systems, alternative representations have been proposed, resulting in the development of so-called matrix product states (MPS). These represent a system of N components or “sites” (corresponding, e.g., to molecular orbitals) by forming products of N matrices, each belonging to one component of the system. The aim of the present project is to develop and implement an efficient quantum chemistry algorithm based on tree tensor network states (QC-TNS), in particular enabling the treatment of problems in quantum chemistry that are intractable by standard techniques as DFT or Coupled Cluster.

What is the significance of the research? Describe the new perspectives opened by the results achieved, including the scientific basics of potential societal applications. Please describe the unique strengths of your proposal in comparison to your domestic and international competitors in the given field. Even though Density Functional Theory (DFT) methods became dominant in the field of electronic structure calculations, there are some special areas where their accuracy is not satisfactory. The applicability of highly accurate multi-reference calculations for such problems is limited to relatively small systems due to their high computational cost. Therefore, computationally less demanding new methods offering ways to accurately compute the electronic correlation are of high importance. In this respect, the QC-TNS, development of which is subject to the current proposal, combines a number of favorable features that suggest it might represent a novel, flexible approach in quantum chemistry: The more general concept of data-sparsity inherent in the TNS representation allows for the efficient representation of a much bigger class of wave functions than accessible by state-of-the-art methods. The desired accuracy may be adjusted, so that the ansatz in principle permeates the whole full-CI space. Additionally, the ansatz is size-consistent by construction and capable of calculating excited states and degenerated eigenfunctions[Legeza-2003b,Dorand-2007]; additional symmetries as particle number and spin symmetries and even non-Abelian symmetries are factored out explicitly [McCulloch-2002,Legeza-2003b,Toth-2008]. The approach might therefore provide a viable alternative in the computation of standard situations, e.g., it may be used to obtain high-precision ab initio reference data for DFT calculations. Further, we are also confident that it offers a solution for the computation of some up to now problematic systems, such as high spin open-shell transition metal complexes [Marti-2008, Marti-2010a, Reiher-2009, Kurashige-2009], where due to heavy elements relativistic effects must be taken into account.

Summary and aims of the research for the public Describe here the major aims of the research for an audience with average background information. This summary is especially important for NKFI in order to inform decision-makers, media, and the taxpayers. The QC-TNS will be built using the experiences and also partially on the structures of two codes, the QC-DMRG-Budapest, designed for MPS computations, and the TTNS-Vienna code, a prototype for tree tensor network computations. The QC-DMRG-Budapest code has first been applied in quantum chemistry application since 2001, and has due to its efficiency recently become the basic tool to carry out large scale DMRG computations in the group of Prof. M. Reiher at ETH Zürich. Among other concepts that ensure fast and stable convergence and optimized use of computational resources, the program features dynamic rank selection (DBSS), optimization of initialization steps (CI-DEAS), partial summation of auxiliary operators, efficient treatment of excited states, treatment of orbital spatial symmetries as well as Abelian and non-Abelian symmetries. The algorithm may be used for the calculation of spin density distributions for molecules that require very large active spaces. Optimization of network structures based on quantum information entropy and orbital of orbital bases are also taken into account to a certain level. These results confirm that the entropy optimized QC-DMRG can provide adequate solutions for difficult problems in transition-metal chemistry, and in fact, it can be used as a ’black box’ method. The TTNS-Vienna code has been developed quite recently and provides a very flexible class structure to handle the more general TNS topology. At present stage it can, however, only be applied to relatively small systems, since many important features related to the QC applications are not implemented yet.

Final report

Results in Hungarian

A kutatás fő célja egy általánosított tenzorhálózat állapot alapú módszer fejlesztése kvantumkémiai rendszerek számára, és a sűrűségmátrix renormálási csoport módszer (DMRG) továbbfejlesztése kvantuminformációelméleti koncepciók felhasználásával. Hogy ezt a célt elérjük, a projektet számos, a kutatási hálózat három csomópontja között megosztott részfeladatra tagoltuk. Ezek magukban foglalták többek közt a fa-tenzorhálózat állapot (TTNS) módszer matematikai kereteinek kifejlesztését, algoritmikus megoldósok implementálását és numerikus szimulációk kivitelezését. Mivel a TTNS számos, a DMRG keretei között kifejlesztett algoritmikus megoldást örököl, először ezeket fejlesztettük és teszteltük a meglévő DMRG kód felhasználásával. Ezen felül, a fejlesztéseink melléktermékeként számos komponenst tudtunk általánosítani és használni nem csak a kvantumkémiában, hanem az erősen korrelált szilárdtestfizikában, ultrahideg atomi rendszerek fizikájában és magfizikában. Kódunkat körülbelül tizenöt kutatócsoport használja világszerte, és a négy éves intervallum alatt 32 munkát publikáltunk, több, mint 15 meghívott előadást mutattunk be nemzetközi konferenciákon, és három kódstruktúrát fejlesztettünk. Elsőként fejlesztettük ki a QCTTNS-t variálható tenzor rend mellett, a négy-komponensű relativisztikus DMRG-t, a DMRG-TCCSD módszert (tailored coupled cluster), fermionikus módustranszformációt és a legkorszerűbb magfizikai DMRG módszert.

Results in English

The main objective of the research was the development of a generalized tensor network state method for quantum chemical systems and further development of the density matrix renormalization group method (DMRG) using concepts of quantum information theory (QIT). In order to achieve such aim, the project has been decomposed into several partial tasks, distributed among the three nodes of the research network. These included among others the development of the mathematical framework of the tree-tensor network state (TTNS) method, implementation of the algorithmic solutions, and performing numerical simulations. Since the TTNS inherits various algorithmic solutions developed in the DMRG framework, first we have developed and tested these using our existing DMRG code. In addition, as a byproduct of our developments, several features could have been generalized and applied not only in quantum chemistry but also in strongly correlated condensed matter and ultra cold atomic systems, and in nuclear structure theory. Our codes are used by some fifteen research groups worldwide, and within the four-year time period we have had 32 publications, presented more than 15 invited talks at international conferences, and developed three code structures. We have developed for the first time the QCTTNS with variable tensor order, the four-component relativistic DMRG, the DMRG-TCCSD (tailored coupled cluster) method, fermionic mode transformation and the state-of-the-art nuclear shell DMRG method.

G. Barcza, Ö. Legeza, R. M. Noack, and J. Sólyom: Entanglement patterns and generalized correlation functions in quantum many body systems, Phys Rev B 92 125140, arXiv:1406.6643, 2015

G Ehlers, J Sólyom, Ö Legeza, R M Noac: Entanglement structure of the Hubbard model in momentum space, Phys Rev B 92 235116 (2015), 2015

Libor Veis, Andrej Antalik, Frank Neese, Ö. Legeza, Jiri Pittner,: Coupled cluster method with single and double excitations tailored by matrix product state wave functions, Journal of Physical Chemistry Letters, 10.1021/acs.jpclett.6b01908, 2016

8. Hagymási I , Legeza Ö: Entanglement, excitations and correlation effects in narrow zigzag graphene nanoribbons, accepted in Phys Rev B arXiv:1605.03041, 2016

Katharina Boguslawski, Paweł Tecmer, Örs Legeza,: Analysis of two-orbital correlations in wavefunctions restricted to electron-pair states, accepted for publication in Phys Rev B, arXiv:1606.08503, 2016

Hagymási I , Legeza Ö: Characterization of a correlated topological Kondo insulator in one dimension, Phys Rev B 93, 165104, 2016

Timár M , Barcza G , Gebhard F , Veis L , Legeza Ö: Hückel-Hubbard-Ohno modeling of π-bonds in ethene and ethyne with application to: Trans -polyacetylene, PHYSICAL CHEMISTRY CHEMICAL PHYSICS 18, 18835, 2016

Katharina Boguslawski, Florent Réal, Paweł Tecmer, Corinne Duperrouzel, André Severo Pereira Gomes, Örs Legeza, Paul W. Ayers, Valérie Vallet,: On the Multi-Reference Nature of Plutonium Oxides: PuO2+2, PuO2, PuO3 and PuO2(OH)2, arXiv:1608.02353 , J Chem Phys, 2016

Szilárd Szalay, Gergely Barcza, Tibor Szilvási, Libor Veis, Örs Legeza,: The correlation theory of the chemical bond, arXiv:1605.06919, 2016

E. Fertitta, D. Koch, B. Paulus, G. Barcza, Ö. Legeza,: Towards a fully size-consistent method of increments, arXiv:1605.03904, 2016

Szilvási T , Barcza G , Legeza Ö: Concept of chemical bond and aromaticity based on quantum information theory, arxiv: submitted to JCTC, 2015

C Krumnow, Ö Legeza, J Eisert: Fermionic orbital optimisation in tensor network states, Phys. Rev. Lett. 117, 210402, Paper arXiv:1504.00042. (2015), 2016

Ö Legeza, L Veis, A Poves, J Dukelsky: Advanced density matrix renormalization group method for nuclear structure calculations, PHYSICAL REVIEW C RAPID COM 92:(5) 051303. 5 p. ArXiv e-prints arXiv:1507.00161: p. 1. (2015), 2015

Yilin Zhao, Katharina Boguslawski, Paweł Tecmer, Corinne Duperrouzel, Gergely Barcza, Örs Legeza, Paul W Ayers: Dissecting the Bond Formation Process of $d^{10}$-Metal-Ethene Complexes with Multireference Approaches, Theoretical Chemistry Accounts 134, 120 2015, 134:120 arXiv:1505.06214 (2015), 2015

Libor Veis, Andrej Antalik, Frank Neese, Ö. Legeza, Jiri Pittner,: Coupled cluster method with single and double excitations tailored by matrix product state wave functions, Journal of Physical Chemistry Letters 7, 4072, 2016, 2016

Katharina Boguslawski, Paweł Tecmer, Örs Legeza: Analysis of two-orbital correlations in wavefunctions restricted to electron-pair states, Phys Rev B 94 155126 (2016), arXiv:1606.08503, 2016

Szilárd Szalay, Gergely Barcza, Tibor Szilvási, Libor Veis, Örs Legeza,: The correlation theory of the chemical bond, Scientific Reports 7, Article number: 2237 (2017) arXiv:1605.06919, 2017

E. Szirmai, G. Barcza, J. Sólyom J, Ö. Legeza: Interplay between exotic superfluidity and magnetism in a chain of four-component ultracold atoms, Phys Rev A 95, 013610 (2017), 2017

K. Boguslawski, F. Réal, P. Tecmer, C. Duperrouzel, A. Serevo, P. Gomes, Ö. Legeza, P.W. Ayers, V. Vallet,: On the Multi-Reference Nature of Plutonium Oxides: PuO2+2, PuO2, PuO3 and PuO2(OH)2, PHYSICAL CHEMISTRY CHEMICAL PHYSICS 19, 4317 (2017, 2017

M. Timár, G. Barcza, F. Gebhard, Ö. Legeza,: Optical phonons for Peierls chains with long-range Coulomb interactions, Phys. Rev B 95, 085150, 2017

Pawel Tecmer, Katharina Boguslawski, Ors Legeza, Markus Reiher: Unravelling the quantum-entanglement effect of noble gas coordination on the spin ground state of CUO, hys. Chem. Chem. Phys, 16, 719–727., 2014, 2014

Örs Legeza, Thorsten Rohwedder, Reinhold Schneider, Szilárd Szalay: Tensor Product Approximation (DMRG) and Coupled Cluster method in Quantum Chemistry, Many-Electron Approaches in Physics, Chemistry and Mathematics (Volker Bach, Luigi Delle Site (Eds.), Springer 2014), 2014

Cs. Nemes, G. Barcza, Z. Nagy, Ö. Legeza, and P. Szolgay: The density matrix renormalization group algorithm on kilo-processor architectures: implementation and trade-offs, Computer Physics Communications 185, 6 (2014), 2014

Stefan Knecht, Ors Legeza, Markus Reiher: Four-Component Density Matrix Renormalization Group, J. Chem. Phys. 140, 041101 (2014), 2014

Matthieu Mottet , Pawel Tecmer, Katharina Boguslawski, Ors Legeza, Markus Reiher: Quantum Entanglement in Carbon-Carbon, Carbon-Phosphorus and Silicon-Silicon Bonds, Phys. Chem. Chem. Phys., 16, 8872-8880 (2014), 2014

V. Murg , F. Verstraete, R. Schneider, P. R. Nagy, O. Legeza: Tree tensor network state study of the ionic-neutral curve crossing of LiF, arXiv:1403.0981, submitted to JCTC, 2014

G. Barcza, Ö. Legeza, R. M. Noack, and J. Sólyom: Entanglement patterns and generalized correlation functions in quantum many body systems, arXiv:1406.6643, Submited to JSTAT, 2014

E. Fertitta, B. Paulus, G. Barcza, and Ö. Legeza: nvestigation of metal- insulator like transition through the ab initio density matrix renormalization group approach, arXiv:1406.7038. Submitted to Phys Rev, 2014

I. Hagymasi , J. Solyom, O. Legeza,: Interorbital interaction in the one- dimensional periodic Anderson model: A density-matrix renormalization- group study, Phys. Rev. B 90, 125137 (2014), 2014

Corinne Duperrouzel, Paweł Tecmer, Katharina Boguslawski, Gergerly Barcza, Örs Legeza, Paul W. Ayers: A quantum informational approach for dissecting chemical reactions, arXiv:1409.4867, Submitted, 2014

V. Murg , F. Verstraete, R. Schneider, P. R. Nagy, O. Legeza: Tree tensor network state study of the ionic-neutral curve crossing of LiF, JOURNAL OF CHEMICAL THEORY AND COMPUTATION 11:(3) pp. 1027-1036. (2015), 2015

G. Barcza, Ö. Legeza, R. M. Noack, and J. Sólyom: Entanglement patterns and generalized correlation functions in quantum many body systems, arXiv:1406.6643, Accepted in PRB, 2014

E. Fertitta, B. Paulus, G. Barcza, and Ö. Legeza: Investigation of metal- insulator like transition through the ab initio density matrix renormalization group approach, arXiv:1406.7038. Accepted in Phys Rev B, 2014

Corinne Duperrouzel, Paweł Tecmer, Katharina Boguslawski, Gergerly Barcza, Örs Legeza, Paul W. Ayers: A quantum informational approach for dissecting chemical reactions, CHEMICAL PHYSICS LETTERS 621: pp. 160-164. (2015), 2015

Barcza G, Szirmai E, Sólyom J, Legeza: Phase separation of superfluids in the chain of four-component ultracold atoms, EUROPEAN PHYSICAL JOURNAL-SPECIAL TOPICS 224:(3) pp. 533-538. (2015), 2015

Brzezicki W, Hagymási I, Dziarmaga J, Legeza O: Second-order Peierls transition in the spin-orbital Kumar-Heisenberg model, PHYSICAL REVIEW B 91:(20) Paper 205137. (2015), 2015

Edoardo Fertitta, Beate Paulus, Gergely Barcza, Örs Legeza: On the calculation of complete dissociation curves of closed-shell pseudo-onedimensional systems through the multireference method of increments, J. Chem. Phys. 143, 114108 (2015), 2015

G Ehlers, J Sólyom, Ö Legeza, R M Noac: Entanglement structure of the Hubbard model in momentum space, arXiv:, 2015

I Hagymasi, J Solyom, O Legeza: Competition between Hund's coupling and Kondo effect in a one-dimensional extended periodic Anderson model, PHYS REV B 92: Paper 035108. (2015), 2015

I Hagymasi, J Solyom, O Legeza: Momentum distribution functions in a one-dimensional extended periodic Anderson model, Advances in Condensed Matter Physics Volume 2015 (2015), Article ID 614017, 2015

C Krumnow, Ö Legeza, J Eisert: Fermionic orbital optimisation in tensor network states, ArXiv e-prints arXiv:1504.00042: Paper arXiv:1504.00042. (2015), 2015

Ö Legeza, L Veis, A Poves, J Dukelsky: Advanced density matrix renormalization group method for nuclear structure calculations, ArXiv e-prints arXiv:1507.00161: p. 1. (2015), 2015

Szalay Sz, Pfeffer M, Murg V, Barcza G, Verstraete F, Schneider R, Legeza Ö: Tensor product methods and entanglement optimization for ab initio quantum chemistry, INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 115:(19) pp. 1342-1391. (2015, 2015

Yilin Zhao, Katharina Boguslawski, Paweł Tecmer, Corinne Duperrouzel, Gergely Barcza, Örs Legeza, Paul W Ayers: Dissecting the Bond Formation Process of $d^{10}$-Metal-Ethene Complexes with Multireference Approaches, Theoretical Chemistry Accounts October 2015, 134:120 arXiv:1505.06214 (2015), 2015