There are a number of interesting and challenging problems that are unsolved yet. At the first instance, these problems may seem simple, but, these problems are floating around for long time. The problem is to describe simple molecules made of the light-weight elements, eg. dimers of H, He, Li, Be, B, C, N, O, F, Ne (not just these: transition-metal dimers also)
Here, I will brief the challenges on each dimer.
- H dimer (H$_{2}$ molecule) : the description of the H$_{2}$ bond breaking (or even more astonishingly H$^{+}$ ion's at various bond lengths) is a difficulty problem to describe by theoretical methods.
- He dimer This has many interesting features. The van der Waals force exists between helium atoms which is the reason for the existence of liquid helium. This is because, at a certain range of distances between atoms the attraction exceeds the repulsion. To understand vdW forces fundamentally, this molecule should be studied. This molecule has other interesting features: 1) He2 is the largest known molecule of two atoms when in its ground state, due to its extremely long bond length. 2) The He2 molecule has a large separation distance between the atoms of about 5200 pm (= 52 ångström). 3) This is the largest for a diatomic molecule without ro-vibronic excitation. 4) The binding energy is only about 1.3 mK, 10−7eV or 1.1×10−5 kcal/mol, or 150 nanoelectron Volts. 5) The bond is 5000 times weaker than the covalent bond in the hydrogen molecule (more on this see this on Wikipedia
- Li dimer This is the lightest metal dimer. (will be updated)
- Be dimer (Be$_{2}$ and Be$_{2+} dimer: It has been difficult to explain the bonding of Be dimer (see this recent Science paper on this topic and this paper using Quantum Monte-Carlo method). Molecular orbital theory, valance band theory, SCF (HF theory), configuration interaction, etc are not able to capture the properties of this elusive molecule. Multi-reference methods also have lead to controversy by predicting different strengths between Be atoms with different bond lengths.
- C dimer (C$_{2}$ molecule) is a difficult problem because of the associated strong static correlation and the difficulty in finding the ground state electron density (see this paper by Ayers and discussion in this paper)
- N dimer (Nitrogen molecule) : The splitting of N$_{2}$ is a difficult problem because of increasing strong static correlation effects as bond is stretched. See this view point by Prof. Burke: "[....its [nitrogen molecule's] triple bond exhibits a high level of static correlation (loosely, electron correlations that arise from the the symmetry of the molecule), which increases as the molecule is stretched. If a computational tool can handle N$_{2}$ well, it can tackle most main-group chemistry correctly."
- O dimer (oxygen molecule): The room temperature O$_{2}$ is largely triplet in ground state and a small portion of the O$_{2}$ are in singlet state. See this JACS article Dioxygen: What Makes This Triplet Diradical Kinetically Persistent?. Also, the dissociation of O$_{2}$ is well studied?
- Transition metal dimers?
Morse potential has been important in understanding molecular spectroscopy of dimers (as well as poly atomic molecules). Here is a Wikipedia text.
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An important extension of the Morse potential that made the Morse form very useful for modern spectroscopy is the MLR (Morse/Long-range) potential.[4] The MLR potential is used as a standard for representing spectroscopic and/or virial data of diatomic molecules by a potential energy curve. It has been used on N2,[5] Ca2,[6] KLi,[7] MgH,[8][9][10] several electronic states of Li2,[4][11][12][13][9][12] Cs2,[14][15] Sr2,[16] ArXe,[9][17] LiCa,[18] LiNa,[19] Br2,[20] Mg2,[21] HF,[22][23] HCl,[22][23] HBr,[22][23] HI,[22][23] MgD,[8] Be2,[24] BeH,[25] and NaH.[26] More sophisticated versions are used for polyatomic molecules.
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While these problems are of interest on the fundamental level, the dimers of transition metal atoms provide applications such as memory storage. Monomer, dimer, trimer on different 2D materials (graphene, phosphorene, etc) are widely studied for their MAE, etc.
(Is there any other dimer which gives challenge to theory or experiment? comment here.
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