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Blasius J, Zaby P, Dölz J, Kirchner B. Uncertainty quantification of phase transition quantities from cluster weighting calculations. J Chem Phys 2022; 157:014505. [DOI: 10.1063/5.0093057] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this work, we investigate how uncertainties in experimental input data influence the results of quantum cluster equilibrium calculations. In particular, we focus on the calculation of vaporization enthalpies and entropies of seven organic liquids, compare two computational approaches for their calculation and investigate how these properties are affected by changes in the experimental input data. It is observed that the vaporization enthalpies and entropies show a smooth dependence on changes in the reference density and boiling point. The reference density is found to have only a small influence on the vaporization thermodynamics, whereas the boiling point has a large influence on the vaporization enthalpy but only a small influence on the vaporization entropy. Furthermore we employed the Gauss--Hermite estimator in order to quantify the uncertainty in the thermodynamic functions that stems from inaccuracies in the experimental reference data at the example of the vaporization enthalpy of (\textit{R})-butan-2-ol. We quantify the uncertainty as 30.95~$\cdot$10$^{-3}$~kJ~mol$^{-1}$. Additionally we compare the convergence behaviour and computational effort of the Gauss--Hermite estimator with the Monte Carlo approach and show the superiority of the former. By this, we present how uncertainty quantification can be applied to examples from theoretical chemistry.
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Affiliation(s)
- Jan Blasius
- University of Bonn Institute of Physical and Theoretical Chemistry, Germany
| | - Paul Zaby
- University of Bonn Institute of Physical and Theoretical Chemistry, Germany
| | - Jürgen Dölz
- Institute for Numerical Simulation, University of Bonn, Friedrich-Hirzebruch-Allee 7 53115 Bonn, Germany, Germany
| | - Barbara Kirchner
- Theoretical and Physical Chemistry, University of Bonn Institute of Physical and Theoretical Chemistry, Germany
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2
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Al-Sheakh L, Fritsch S, Appelhagen A, Villinger A, Ludwig R. Thermodynamically Stable Cationic Dimers in Carboxyl-Functionalized Ionic Liquids: The Paradoxical Case of "Anti-Electrostatic" Hydrogen Bonding. Molecules 2022; 27:molecules27020366. [PMID: 35056680 PMCID: PMC8778807 DOI: 10.3390/molecules27020366] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2021] [Revised: 12/28/2021] [Accepted: 01/03/2022] [Indexed: 11/18/2022] Open
Abstract
We show that carboxyl-functionalized ionic liquids (ILs) form doubly hydrogen-bonded cationic dimers (c+=c+) despite the repulsive forces between ions of like charge and competing hydrogen bonds between cation and anion (c+–a−). This structural motif as known for formic acid, the archetype of double hydrogen bridges, is present in the solid state of the IL 1−(carboxymethyl)pyridinium bis(trifluoromethylsulfonyl)imide [HOOC−CH2−py][NTf2]. By means of quantum chemical calculations, we explored different hydrogen-bonded isomers of neutral (HOOC–(CH2)n–py+)2(NTf2−)2, single-charged (HOOC–(CH2)n–py+)2(NTf2−), and double-charged (HOOC– (CH2)n−py+)2 complexes for demonstrating the paradoxical case of “anti-electrostatic” hydrogen bonding (AEHB) between ions of like charge. For the pure doubly hydrogen-bonded cationic dimers (HOOC– (CH2)n−py+)2, we report robust kinetic stability for n = 1–4. At n = 5, hydrogen bonding and dispersion fully compensate for the repulsive Coulomb forces between the cations, allowing for the quantification of the two equivalent hydrogen bonds and dispersion interaction in the order of 58.5 and 11 kJmol−1, respectively. For n = 6–8, we calculated negative free energies for temperatures below 47, 80, and 114 K, respectively. Quantum cluster equilibrium (QCE) theory predicts the equilibria between cationic monomers and dimers by considering the intermolecular interaction between the species, leading to thermodynamic stability at even higher temperatures. We rationalize the H-bond characteristics of the cationic dimers by the natural bond orbital (NBO) approach, emphasizing the strong correlation between NBO-based and spectroscopic descriptors, such as NMR chemical shifts and vibrational frequencies.
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Affiliation(s)
- Loai Al-Sheakh
- Institut für Chemie, Abteilung für Physikalische Chemie, Universität Rostock, Dr.-Lorenz-Weg 2, 18059 Rostock, Germany; (L.A.-S.); (S.F.); (A.A.)
| | - Sebastian Fritsch
- Institut für Chemie, Abteilung für Physikalische Chemie, Universität Rostock, Dr.-Lorenz-Weg 2, 18059 Rostock, Germany; (L.A.-S.); (S.F.); (A.A.)
| | - Andreas Appelhagen
- Institut für Chemie, Abteilung für Physikalische Chemie, Universität Rostock, Dr.-Lorenz-Weg 2, 18059 Rostock, Germany; (L.A.-S.); (S.F.); (A.A.)
| | - Alexander Villinger
- Institut für Chemie, Abteilung für Anorganische Chemie, Universität Rostock, Albert-Einstein-Str. 3a, 18059 Rostock, Germany;
| | - Ralf Ludwig
- Institut für Chemie, Abteilung für Physikalische Chemie, Universität Rostock, Dr.-Lorenz-Weg 2, 18059 Rostock, Germany; (L.A.-S.); (S.F.); (A.A.)
- Department LL&M, University of Rostock, Albert-Einstein−Str. 25, 18059 Rostock, Germany
- Leibniz−Institut für Katalyse an der Universität Rostock e.V., Albert-Einstein−Str. 29a, 18059 Rostock, Germany
- Correspondence:
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3
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Zaby P, Ingenmey J, Kirchner B, Grimme S, Ehlert S. Calculation of improved enthalpy and entropy of vaporization by a modified partition function in quantum cluster equilibrium theory. J Chem Phys 2021; 155:104101. [PMID: 34525835 DOI: 10.1063/5.0061187] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
Abstract
In this work, we present an altered partition function that leads to an improved calculation of the enthalpy and entropy of vaporization in the framework of quantum cluster equilibrium theory. The changes are based on a previously suggested modification [S. Grimme, Chem. Eur. J. 18, 9955-9964 (2012)] of the molecular entropy calculation in the gas phase. Here, the low energy vibrational frequencies in the vibrational partition function are treated as hindered rotations instead of vibrations. The new scheme is tested on a set of nine organic solvents for the calculation of the enthalpy and entropy of vaporization. The enthalpies and entropies of vaporization show improvements from 6.5 error to 3.3 kJ mol-1 deviation to experiment and from 28.4 error to 13.5 J mol-1 K-1 deviation to experiment, respectively. The effect of the corrected partition function is visible in the different populations of clusters, which become physically more meaningful in that larger clusters are higher populated in the liquid phase and the gas phase is mainly populated by the monomers. Furthermore, the corrected partition function also overcomes technical difficulties and leads to an increased stability of the calculations in regard to the size of the cluster set.
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Affiliation(s)
- Paul Zaby
- Mulliken Center for Theoretical Chemistry, University of Bonn, Bonn, Germany
| | - Johannes Ingenmey
- Mulliken Center for Theoretical Chemistry, University of Bonn, Bonn, Germany
| | - Barbara Kirchner
- Mulliken Center for Theoretical Chemistry, University of Bonn, Bonn, Germany
| | - Stefan Grimme
- Mulliken Center for Theoretical Chemistry, University of Bonn, Bonn, Germany
| | - Sebastian Ehlert
- Mulliken Center for Theoretical Chemistry, University of Bonn, Bonn, Germany
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Perlt E, Berger SA, Kelterer AM, Kirchner B. Anharmonicity of Vibrational Modes in Hydrogen Chloride-Water Mixtures. J Chem Theory Comput 2019; 15:2535-2547. [PMID: 30811198 DOI: 10.1021/acs.jctc.8b01070] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
A thorough analysis of molecular vibrations in the binary system hydrogen chloride/water is presented considering a set of small mixed and pure clusters. In addition to the conventional normal-mode analysis based on the diagonalization of the Hessian, anharmonic frequencies were obtained from the perturbative VPT2 and PT2-VSCF method using hybrid density functional theory. For all normal modes, potential energy curves were modeled by displacing the atoms from the minimum geometry along the normal mode vectors. Three model potentials, a harmonic potential, a Morse potential, and a fourth order polynomial, were applied to fit these curves. From these data, it was possible not only to characterize distinct vibrations as mainly harmonic, anharmonic, or involving higher order terms but also to extract force constants, k, and anharmonicity constants, xe. By investigating all different types of intramolecular vibrations including covalent stretching or bending vibrations and intermolecular vibrations such as librations, we could demonstrate that while vibrational frequencies can be obtained applying scaling factors to harmonic results, useful anharmonicity constants cannot be predicted in such a way and the usage of more elaborate vibrational methods is necessary. For each particular type of molecular vibration, we could however determine a relationship between the wavenumber or wavenumber shift and the anharmonicity constant, which allows us to estimate mode dependent anharmonicity constants for larger clusters in the future.
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Affiliation(s)
- Eva Perlt
- Department of Chemistry , University of California, Irvine , 1102 Natural Sciences II , Irvine , California 92697-2025 , United States
| | - Sarah A Berger
- Institute of Physical and Theoretical Chemistry, NAWI Graz , Graz University of Technology , Stremayrgasse 9 , 8010 Graz , Austria
| | - Anne-Marie Kelterer
- Institute of Physical and Theoretical Chemistry, NAWI Graz , Graz University of Technology , Stremayrgasse 9 , 8010 Graz , Austria
| | - Barbara Kirchner
- Mulliken Center for Theoretical Chemistry , University of Bonn , Beringstrasse 4 , D-53115 Bonn , Germany
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Izato YI, Matsugi A, Koshi M, Miyake A. A simple heuristic approach to estimate the thermochemistry of condensed-phase molecules based on the polarizable continuum model. Phys Chem Chem Phys 2019; 21:18920-18929. [DOI: 10.1039/c9cp03226f] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
A simple model based on a quantum chemical approach with polarizable continuum models (PCMs) to provide reasonable translational and rotational entropies for liquid phase molecules was developed.
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Affiliation(s)
- Yu-ichiro Izato
- Graduate School of Information and Environment Sciences
- Yokohama National University
- Yokohama
- Japan
| | - Akira Matsugi
- National Institute of Advanced Industrial Sciences and Technology
- Ibaraki
- Japan
| | - Mitsuo Koshi
- Department of Chemical System Engineering
- The University of Tokyo
- Tokyo
- Japan
| | - Atsumi Miyake
- Institute of Advanced Sciences
- Yokohama National University
- Yokohama
- Japan
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Seeger ZL, Kobayashi R, Izgorodina EI. Cluster approach to the prediction of thermodynamic and transport properties of ionic liquids. J Chem Phys 2018; 148:193832. [DOI: 10.1063/1.5009791] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022] Open
Affiliation(s)
- Zoe L. Seeger
- School of Chemistry, Monash University, 17 Rainforest Walk, Clayton, VIC 3800, Australia
| | - Rika Kobayashi
- Australian National University, Leonard Huxley Building 56, Mills Road, Canberra, ACT 2601, Australia
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Abstract
We present a first-principles calculation and mechanistic characterization of the ion product of liquid water (K W ), based on Quantum Cluster Equilibrium (QCE) theory with a variety of ab initio and density functional methods. The QCE method is based on T-dependent Boltzmann weighting of different-sized clusters and consequently enables the observation of thermodynamically less favored and therefore low populated species such as hydronium and hydroxide ions in water. We find that common quantum chemical methods achieve semi-quantitative accuracy in predicting K W and its T-dependence. Dominant ion-pair water clusters of the QCE equilibrium distribution are found to exhibit stable 2-coordinate buttress-type motifs, all with maximally Grotthus-ordered H-bond patterns that successfully prevent recombination of hydronium and hydroxide ions at 3-coordinate bridgehead sites. We employ standard quantum chemistry techniques to describe kinetic and mechanistic aspects of ion-pair formation, and we obtain NBO-based bonding indices to characterize other electronic, structural, spectroscopic, and reactive properties of cluster-mediated ionic dissociation.
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Ingenmey J, von Domaros M, Kirchner B. Predicting miscibility of binary liquids from small cluster QCE calculations. J Chem Phys 2017; 146:154502. [PMID: 28433040 DOI: 10.1063/1.4980032] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Affiliation(s)
- Johannes Ingenmey
- Mulliken Center for Theoretical Chemistry, Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany
| | - Michael von Domaros
- Mulliken Center for Theoretical Chemistry, Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany
| | - Barbara Kirchner
- Mulliken Center for Theoretical Chemistry, Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany
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von Domaros M, Perlt E. Anharmonic effects in the quantum cluster equilibrium method. J Chem Phys 2017; 146:124114. [PMID: 28388115 DOI: 10.1063/1.4978958] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The well-established quantum cluster equilibrium (QCE) model provides a statistical thermodynamic framework to apply high-level ab initio calculations of finite cluster structures to macroscopic liquid phases using the partition function. So far, the harmonic approximation has been applied throughout the calculations. In this article, we apply an important correction in the evaluation of the one-particle partition function and account for anharmonicity. Therefore, we implemented an analytical approximation to the Morse partition function and the derivatives of its logarithm with respect to temperature, which are required for the evaluation of thermodynamic quantities. This anharmonic QCE approach has been applied to liquid hydrogen chloride and cluster distributions, and the molar volume, the volumetric thermal expansion coefficient, and the isobaric heat capacity have been calculated. An improved description for all properties is observed if anharmonic effects are considered.
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Affiliation(s)
- Michael von Domaros
- Mulliken Center for Theoretical Chemistry, University of Bonn, Beringstr. 6, D-53115 Bonn, Germany
| | - Eva Perlt
- Mulliken Center for Theoretical Chemistry, University of Bonn, Beringstr. 6, D-53115 Bonn, Germany
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Izgorodina EI, Seeger ZL, Scarborough DLA, Tan SYS. Quantum Chemical Methods for the Prediction of Energetic, Physical, and Spectroscopic Properties of Ionic Liquids. Chem Rev 2017; 117:6696-6754. [PMID: 28139908 DOI: 10.1021/acs.chemrev.6b00528] [Citation(s) in RCA: 125] [Impact Index Per Article: 17.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
Abstract
The accurate prediction of physicochemical properties of condensed systems is a longstanding goal of theoretical (quantum) chemistry. Ionic liquids comprising entirely of ions provide a unique challenge in this respect due to the diverse chemical nature of available ions and the complex interplay of intermolecular interactions among them, thus resulting in the wide variability of physicochemical properties, such as thermodynamic, transport, and spectroscopic properties. It is well understood that intermolecular forces are directly linked to physicochemical properties of condensed systems, and therefore, an understanding of this relationship would greatly aid in the design and synthesis of functionalized materials with tailored properties for an application at hand. This review aims to give an overview of how electronic structure properties obtained from quantum chemical methods such as interaction/binding energy and its fundamental components, dipole moment, polarizability, and orbital energies, can help shed light on the energetic, physical, and spectroscopic properties of semi-Coulomb systems such as ionic liquids. Particular emphasis is given to the prediction of their thermodynamic, transport, spectroscopic, and solubilizing properties.
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Affiliation(s)
- Ekaterina I Izgorodina
- Monash Computational Chemistry Group, School of Chemistry, Monash University , 17 Rainforest Walk, Clayton, Victoria 3800, Australia
| | - Zoe L Seeger
- Monash Computational Chemistry Group, School of Chemistry, Monash University , 17 Rainforest Walk, Clayton, Victoria 3800, Australia
| | - David L A Scarborough
- Monash Computational Chemistry Group, School of Chemistry, Monash University , 17 Rainforest Walk, Clayton, Victoria 3800, Australia
| | - Samuel Y S Tan
- Monash Computational Chemistry Group, School of Chemistry, Monash University , 17 Rainforest Walk, Clayton, Victoria 3800, Australia
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von Domaros M, Jähnigen S, Friedrich J, Kirchner B. Quantum cluster equilibrium model of N-methylformamide–water binary mixtures. J Chem Phys 2016; 144:064305. [DOI: 10.1063/1.4941278] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Michael von Domaros
- Mulliken Center for Theoretical Chemistry, Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany
| | - Sascha Jähnigen
- Martin-Luther-Universität Halle-Wittenberg, von-Danckelmann-Platz 4, D-06120 Halle, Germany
| | - Joachim Friedrich
- Technische Universität Chemnitz, Straße der Nationen 62, D-09111 Chemnitz, Germany
| | - Barbara Kirchner
- Mulliken Center for Theoretical Chemistry, Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany
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12
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Pensado AS, Brehm M, Thar J, Seitsonen AP, Kirchner B. Effect of Dispersion on the Structure and Dynamics of the Ionic Liquid 1-Ethyl-3-methylimidazolium Thiocyanate. Chemphyschem 2012; 13:1845-53. [DOI: 10.1002/cphc.201100917] [Citation(s) in RCA: 76] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2011] [Indexed: 11/09/2022]
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13
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Brüssel M, Perlt E, Lehmann SBC, von Domaros M, Kirchner B. Binary systems from quantum cluster equilibrium theory. J Chem Phys 2011; 135:194113. [DOI: 10.1063/1.3662071] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023] Open
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14
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Spickermann C, Perlt E, von Domaros M, Roatsch M, Friedrich J, Kirchner B. Coupled Cluster in Condensed Phase. Part II: Liquid Hydrogen Fluoride from Quantum Cluster Equilibrium Theory. J Chem Theory Comput 2011; 7:868-75. [DOI: 10.1021/ct200074c] [Citation(s) in RCA: 31] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022]
Affiliation(s)
- Christian Spickermann
- Wilhelm-Ostwald-Institut für Physikalische und Theoretische Chemie, Universität Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
| | - Eva Perlt
- Wilhelm-Ostwald-Institut für Physikalische und Theoretische Chemie, Universität Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
| | - Michael von Domaros
- Wilhelm-Ostwald-Institut für Physikalische und Theoretische Chemie, Universität Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
| | - Martin Roatsch
- Wilhelm-Ostwald-Institut für Physikalische und Theoretische Chemie, Universität Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
| | - Joachim Friedrich
- Institute for Chemistry, Chemnitz University of Technology, Strasse der Nationen 62, 09111 Chemnitz, Germany
| | - Barbara Kirchner
- Wilhelm-Ostwald-Institut für Physikalische und Theoretische Chemie, Universität Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
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Abstract
First, different approaches to detect hydrogen bonds and to evaluate their energies are introduced newly or are extended. Supermolecular interaction energies of 256 dimers, each containing one single hydrogen bond, were correlated to various descriptors by a fit function depending both on the donor and acceptor atoms of the hydrogen bond. On the one hand, descriptors were orbital-based parameters as the two-center or three-center shared electron number, products of ionization potentials and shared electron numbers, and the natural bond orbital interaction energy. On the other hand, integral descriptors examined were the acceptor-proton distance, the hydrogen bond angle, and the IR frequency shift of the donor-proton stretching vibration. Whereas an interaction energy dependence on 1/r(3.8) was established, no correlation was found for the angle. Second, the fit functions are applied to hydrogen bonds in polypeptides, amino acid dimers, and water cluster, thus their reliability is demonstrated. Employing the fit functions to assign intramolecular hydrogen bond energies in polypeptides, several side chain CH...O and CH...N hydrogen bonds were detected on the fly. Also, the fit functions describe rather well intermolecular hydrogen bonds in amino acid dimers and the cooperativity of hydrogen bond energies in water clusters.
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Affiliation(s)
- Katharina Wendler
- Lehrstuhl für Theoretische Chemie, Wilhelm-Ostwald-Institut für Physikalische und Theoretische Chemie, Universität Leipzig, Germany
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Lehmann SBC, Spickermann C, Kirchner B. Quantum Cluster Equilibrium Theory Applied in Hydrogen Bond Number Studies of Water. 2. Icebergs in a Two-Dimensional Water Continuum? J Chem Theory Comput 2009; 5:1650-6. [DOI: 10.1021/ct900189v] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- S. B. C. Lehmann
- Wilhelm-Ostwald Institute of Physical and Theoretical Chemistry, University of Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
| | - C. Spickermann
- Wilhelm-Ostwald Institute of Physical and Theoretical Chemistry, University of Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
| | - B. Kirchner
- Wilhelm-Ostwald Institute of Physical and Theoretical Chemistry, University of Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
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Lehmann SBC, Spickermann C, Kirchner B. Quantum Cluster Equilibrium Theory Applied in Hydrogen Bond Number Studies of Water. 1. Assessment of the Quantum Cluster Equilibrium Model for Liquid Water. J Chem Theory Comput 2009; 5:1640-9. [DOI: 10.1021/ct800310a] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- S. B. C. Lehmann
- Wilhelm-Ostwald Institute of Physical and Theoretical Chemistry, University of Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
| | - C. Spickermann
- Wilhelm-Ostwald Institute of Physical and Theoretical Chemistry, University of Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
| | - B. Kirchner
- Wilhelm-Ostwald Institute of Physical and Theoretical Chemistry, University of Leipzig, Linnéstrasse 2, D-04103 Leipzig, Germany
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